tag:blogger.com,1999:blog-38321360178937494972017-04-26T09:50:40.164-04:00Elliptic ComposabilityFlorin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.comBlogger213125tag:blogger.com,1999:blog-3832136017893749497.post-63379044842014083972017-04-23T20:52:00.000-04:002017-04-23T20:52:04.238-04:00<h2 style="text-align: center;">Politics and a bit on the symmetry properties of the commutator and the Jordan products</h2><div><br /></div><div>This week I am thorn between a physics and politics. On one hand I have the scheduled physics topics to talk about, and on the other hand there are very juicy political topics. So let me start with some political commentary which I will attempt to keep at the minimum.</div><div><br /></div><div>If you do not live in US, it is hard to understand the amount of political pressure on science which comes from the right. GOP is at war with science because of three factors: the political elites are corrupt and depend on lobby money from corporations who make more money when they wreck the environment, the religious right is at war with evolution, and lastly the inbred rednecks swallow hook, line, and sinker the toxic sludge of propaganda of Fox News in the name of "freedom". </div><div><br /></div><div>So it was a breath of fresh air the recent march for science in which people sick and tired of the GOP war on science took a stand for the facts that 2+2 is still 4, humans cause global warming, and Earth is older than 5000 years. And then I opened my email and I see an alert about a new post by Lubos Motl defending Bill O'Reilly. I normally delete those notifications and I don't really know how I am subscribed to them because I only get about one a week - it is strangely inconsistent. So I said to myself: how fitting. The (naked) emperor of physics who wanted once to reclassify an archive paper to the general physics section <b>after it was published in PRL, </b>the climate change denier and the open apologist of the murderer Putin, con-man Trump, and white trash Sarah Palin throws his support behind the another toxic sorry propagandist like himself. Would have been too much to expect him to defend science instead? Out of curiosity I followed the link to see the pro O'Reilly rant, and I was not disappointed: it was choke full of imbecilic nonsense as I expected. But then I saw the icing on the cake: I see in the history list that Lubos did write a rant against the march for science too calling it misguided and unethical. Wow! Now in France (like in US or UK) there is no shortage of stupidity which just propelled Marine LePen into the final for presidency. The global village idiots will flock to her side and I have no doubt Lubos will support her too.</div><div><br /></div><div>OK, the political topics took too much and I want to continue with the series topic on quantum mechanics reconstruction. Let me just say what the products \(\alpha\) and \(\sigma\) will turn out to be. In the classical mechanics case it can be constructively proven that \(\alpha\) is the Poisson bracket while in the quantum case, \(\alpha\) is the commutator. The other product \(\sigma\) is the regular function multiplication in classical mechanics and the Jordan product (the anti-commutator) in quantum mechanics. </div><div><br /></div><div>Now the Poisson bracket and the commutators are anti-symmetric: \(f\alpha g = - g\alpha f\), and the regular function multiplication and the Jordan product are symmetric products: \(f\sigma g = g\sigma f\). The symmetry properties are preserved under system composition as we can see from the fundamental relationships:<br /><br />\(\Delta (\alpha) = \alpha \otimes \sigma + \sigma \otimes \alpha \)<br />\(\Delta (\sigma) = \sigma \otimes \sigma - \alpha \otimes \alpha\)<br /><br />because S*S = S, S*A = A, A*A = S<br /><br />Incidentally, this observation opens up another way into quantum mechanics reconstruction (from the operational point of view) but I will not talk about it in this series. Instead next time I will show how to prove the fact that the product \(\alpha\) is anti-symmetric. Again Leibniz identity will come to the rescue. Then using the fundamental relationship we must have that the product \(\sigma\) is symmetric. Eventually all their mathematical properties will be obtained. Please stay tuned. </div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-28911436502100318092017-04-16T23:52:00.001-04:002017-04-16T23:52:31.915-04:00<h2 style="text-align: center;">The fundamental bipartite relations</h2><div><br /></div><div>Continuing from where we left off last time, we introduced the most general composite products for a bipartite system:</div><div><br /></div><div><div>\(\alpha_{12} = a_{11}\alpha \otimes \alpha + a_{12} \alpha\otimes\sigma + a_{21} \sigma\otimes \alpha + a_{22} \sigma\otimes\sigma\)</div><div>\(\sigma_{12} = b_{11}\alpha \otimes \alpha + b_{12} \alpha\otimes\sigma + b_{21} \sigma\otimes \alpha + b_{22} \sigma\otimes\sigma\)<br /><br />The question now becomes: are the \(a\)'s and \(b\)'s parameters free, or can we say something abut them? To start let's normalize the products \(\sigma\) like this:<br /><br />\(f\sigma I = I\sigma f = f\)<br /><br />which can always be done. Now in:<br /><br /><div>\((f_1 \otimes f_2)\alpha_{12}(g_1\otimes g_2) = \)</div><div>\(=a_{11}(f_1 \alpha g_1)\otimes (f_2 \alpha g_2) + a_{12}(f_1 \alpha g_1) \otimes (f_2 \sigma g_2 ) +\)</div><div>\(+a_{21}(f_1 \sigma g_1)\otimes (f_2 \alpha g_2) + a_{22}(f_1 \sigma g_1) \otimes (f_2 \sigma g_2 )\)<br /><br />if we pick \(f_1 = g_1 = I\) :<br /><br /><div>\((I \otimes f_2)\alpha_{12}(I\otimes g_2) = \)</div><div>\(=a_{11}(I \alpha I)\otimes (f_2 \alpha g_2) + a_{12}(I \alpha I) \otimes (f_2 \sigma g_2 ) +\)</div><div>\(+a_{21}(I \sigma I)\otimes (f_2 \alpha g_2) + a_{22}(I \sigma I) \otimes (f_2 \sigma g_2 )\)<br /><br />and recalling from last time that \(I\alpha I = 0\) from Leibniz identity we get:<br /><br />\(f_2 \alpha g_2 = a_{21} (f_2 \alpha g_2 ) + a_{22} (f_2 \sigma g_2)\)<br /><br />which demands \(a_{21} = 1\) and \(a_{22} = 0\).<br /><br />If we make the same substitution into:<br /><br /> \((f_1 \otimes f_2)\sigma_{12}(g_1\otimes g_2) = \)<br /><div>\(=b_{11}(f_1 \alpha g_1)\otimes (f_2 \alpha g_2) + b_{12}(f_1 \alpha g_1) \otimes (f_2 \sigma g_2 ) +\)</div><div>\(+b_{21}(f_1 \sigma g_1)\otimes (f_2 \alpha g_2) + b_{22}(f_1 \sigma g_1) \otimes (f_2 \sigma g_2 )\)<br /><br />we get:<br /><br />\(f_2 \sigma g_2 = b_{21} (f_2 \alpha g_2 ) + b_{22} (f_2 \sigma g_2)\)<br /><br />which demands \(b_{21} = 0\) and \(b_{22} = 1\)<br /><br />We can play the same game with \(f_2 = g_2 = I\) and (skipping the trivial details) we get two additional conditions: \(a_{12} = 1\) and \(b_{12} = 0\).<br /><br />In coproduct notation what we get so far is:<br /><br />\(\Delta (\alpha) = \alpha \otimes \sigma + \sigma \otimes \alpha + a_{11} \alpha \otimes \alpha\)<br />\(\Delta (\sigma) = \sigma \otimes \sigma + b_{11} \alpha \otimes \alpha\)<br /><br />By applying Leibniz identity on a bipartite system, one can show after some tedious computations that \(a_{11} = 0\). The only remaining free parameters is \(b_{11}\) which can be normalized to be ether -1, 0, or 1 (or elliptic, parabolic, and hyperbolic). <b>Each choice corresponds to a potential theory of nature</b>. For example 0 corresponds to classical mechanics, and -1 to quantum mechanics.<br /><br /><b>Elliptic composability is quantum mechanics! The bipartite products obey:</b><br /><b><br /></b><b><br /></b>\(\Delta (\alpha) = \alpha \otimes \sigma + \sigma \otimes \alpha \)<br />\(\Delta (\sigma) = \sigma \otimes \sigma - \alpha \otimes \alpha\)<br /><br /><b>Please notice the similarity with complex number multiplication. This is why complex numbers play a central role in quantum mechanics.</b><br /><b><br /></b>Now at the moment the two products do not respect any other properties. But we can continue this line of argument and prove their symmetry/anti-symmetry. And from there we can derive their complete properties arriving constructively at the standard formulation of quantum mechanics. Please stay tuned.</div></div></div></div></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-75737353711244180922017-04-09T20:49:00.001-04:002017-04-09T20:49:47.734-04:00<h2 style="text-align: center;">Time evolution for a composite system</h2><div><br /></div><div>Continuing where we left off last time, let me first point out one thing which I glossed over too fast: the representation of \(D\) as a product \(\alpha\): \(Dg = f\alpha g\). This is highly nontrivial and <b>not all time evolutions respect it</b>. In fact, the statement above is nothing but a reformulation of <a href="https://en.wikipedia.org/wiki/Noether%27s_theorem" target="_blank">Noether's theorem</a> in the Hamiltonian formalism. I did not build up the proper mathematical machinery to easily show this, so take my word on it for now. I might revisit this at a later time.</div><div><br /></div><div>Now what I want to do is explore what happens to the product \(\alpha\) when we consider two physical systems 1 and 2. First, let's introduce the unit element of our category, and let's call it "I":</div><div><br /></div><div>\(f\otimes I = I\otimes f = f\)</div><div><br /></div><div>for all \(f \in C\)</div><div><br /></div><div>Then we have \((f_1\otimes I) \alpha_{12} (g_1\otimes I) = f \alpha g\)</div><div><br /></div><div>On the other hand suppose in nature there exists only the product \(\alpha\). Then the only way we can construct a composite product \(\alpha_{12}\) out of \(\alpha_1\) and \(\alpha_2\) is:</div><div><br /></div><div>\((f_1\otimes f_2) \alpha_{12} (g_1 \otimes g_2) = a(f_1 \alpha_1 g_1)\otimes (f_2\alpha_2 g_2)\)</div><div><br /></div><div>where \(a\) is a constant. </div><div><br /></div><div>Now if we pick \(f_2 = g_2 = I\) we get:</div><div><br /></div><div>\((f_1\otimes I) \alpha_{12} (g_1 \otimes I) = a(f_1 \alpha_1 g_1)\otimes (I \alpha_2 I) \)</div><div>which is the same as \(f \alpha g\) by above. </div><div><br /></div><div>But what is \(I\alpha I\)? Here we use the Leibniz identity and prove it is equal with zero:</div><div><br /></div><div>\(I \alpha (I\alpha A) = (I \alpha I) \alpha A + I \alpha (I \alpha A)\)</div><div><br /></div><div>for all \(A\) and hence \(I\alpha I = 0\)</div><div><br /></div><div>But this means that <b>a single product alpha by itself is not enough! Therefore we need a second product </b>\(\sigma\)! Alpha will turn out to be the commutator, and sigma the Jordan product of observables, but we will derive this in a constructive fashion.</div><div><br /></div><div>Now that we have two products in our theory of nature, let's see how can we build the composite products out of individual systems. Basically we try all possible combinations:</div><div><br /></div><div>\(\alpha_{12} = a_{11}\alpha \otimes \alpha + a_{12} \alpha\otimes\sigma + a_{21} \sigma\otimes \alpha + a_{22} \sigma\otimes\sigma\)</div><div>\(\sigma_{12} = b_{11}\alpha \otimes \alpha + b_{12} \alpha\otimes\sigma + b_{21} \sigma\otimes \alpha + b_{22} \sigma\otimes\sigma\)</div><div><br /></div><div>which is shorthand for (I am spelling out only the first case):</div><div><br /></div><div>\((f_1 \otimes f_2)\alpha_{12}(g_1\otimes g_2) = \)</div><div>\(=a_{11}(f_1 \alpha g_1)\otimes (f_2 \alpha g_2) + a_{12}(f_1 \alpha g_1) \otimes (f_2 \sigma g_2 ) +\)</div><div>\(+a_{21}(f_1 \sigma g_1)\otimes (f_2 \alpha g_2) + a_{22}(f_1 \sigma g_1) \otimes (f_2 \sigma g_2 )\)</div><div><br /></div><div>For the mathematically inclined reader we have constructed what it is called a <a href="https://en.wikipedia.org/wiki/Coalgebra" target="_blank">coalgebra </a>where the operation is called a coproduct: \(\Delta : C \rightarrow C\otimes C\). <b>In category theory a coproduct is obtained from a product by reversing the arrows.</b></div><div><b><br /></b></div><div>Now the task is to see if we can say something about the coproduct parameters: \(a_{11},..., b_{22}\). In general nothing can constrain their values, but in our case we do have an additional relation: Leibniz identity which arises out the functoriality of time evolution. This will be enough to fully determine the products \(\alpha\) and \(\sigma\), and from them the formalism of quantum mechanics. Please stay tuned.</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-17661262499437536722017-03-26T18:30:00.000-04:002017-04-09T17:53:59.273-04:00<h2 style="text-align: center;">Time as a continous functor</h2><div><br /></div><div>To recall from prior posts, a functor maps objects to objects and arrows to arrows between two categories. In other words, it is structure preserving. In the case of a monoidal category, suppose there is an arrow * from \(C\times C \rightarrow C\). Then a functor T makes the diagram below commute:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-CUUpUxprXSA/WNgTFMIrt-I/AAAAAAAABLA/PwSgLfe0WyckeG022zqPP7r13Fg0HMEfgCLcB/s1600/time.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="https://3.bp.blogspot.com/-CUUpUxprXSA/WNgTFMIrt-I/AAAAAAAABLA/PwSgLfe0WyckeG022zqPP7r13Fg0HMEfgCLcB/s320/time.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">This is all fancy abstract math which has a simple physical interpretation when T corresponds to time evolution: <b>the laws of physics do not change in time. </b>Moreover it can be shown with a bit of effort and knowledge of C* algebras that <b>Time as a functor = unitarity</b>.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">But what can we derive from the commutative diagram above? With the additional help of two more very simple and natural ingredients we will be able to reconstruct the complete formalism of quantum mechanics!!! Today I will introduce the first one: time is a continuous parameter. Just like in group theory adding continuity results in the theory of Lie groups we will consider <b><u>continous functors</u> and we will investigate what happens in the neighborhood of the identity element.</b></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">In the limit of time evolution going to zero T becomes the identity. For infinitesimal time evolution we can then write:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">\(T = I + \epsilon D\)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">We plug this back into the diagram commutativity condition \(T(A)*T(B) = T(A*B)\) and we obtain in first order the chain rule of differentiation:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">\(D(A*B) = D(A)*B + A*D(B)\)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">There is not a single kind of time evolution and \(D\) is not unique (think of various hamiltonians). There is a natural transformation between different time evolution functors and we can express D as an operation like this: \(D_A = A\alpha\) where \((\cdot \alpha \cdot)\) is a product.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">\(\alpha : C\times C \rightarrow C\)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Then we obtain the Leibniz identity:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">\(A\alpha (B * C) = (A\alpha B) * C + B * (A \alpha C)\)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This is extremely powerful, as it is unitarity in disguise. Next time we'll use the tensor product and the second ingredient to obtain many more mathematical consequences. Please stay tuned.</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-88643292734778322672017-03-19T19:13:00.000-04:002017-03-19T19:13:50.992-04:00<h2 style="text-align: center;">Monoidal categories and the tensor product</h2><div><br /></div><div><br /></div><div>Last time we discussed the category theory product which forms another category from two categories. Suppose now that we start with one category \(C\) and form the product with itself \(C\times C\). It is natural to see if there is a functor from \(C\times C\) to \(C\). If such a functor exists and moreover it respects associativity and unit elements, then the category \(C\) is called a <b>monoidal category. </b>By abuse of notation, the functor above is called the tensor product, but this is not the usual tensor product of vector space. The tensor product of vector space is only one concrete example of a monoidal product. To get to the ordinary tensor product we need to inject physics into the problem. </div><div><br /></div><div>The category \(C\) we are interested in is that of physical systems where the objects are physical systems, and arrows are compositions of physical systems. The key physical concepts needed are that of <b>time</b> and <b>dynamical degree of freedom </b>inside <b>Hamiltonian formalism.</b></div><div><br /></div><div>Time plays an distinguished role in quantum mechanics both in terms of formalism (remember that there is no time operator) and in how quantum mechanics can be reconstructed. </div><div><br /></div><div>The space in Hamiltonian formalism is a Poisson manifold which is not necessarily a vector space but because the Hilbert space \(L^2 (R^3\times R^3)\) is isomorphic to \(L^2 (R^3 ) \otimes L^2 (R^3 )\) let's discuss monoidal categories for vector spaces obeying an equivalence relationship. Hilbert spaces form a category of their own and there is a functor mapping physical systems into Hilbert spaces. This is usually presented as the first quantum mechanics postulate: each physical system is associated with a complex Hilbert space H.</div><div><br /></div><div>For complete generality of the definition of the tensor product we consider two distinct vector space V and W for which we first consider the category theory product (in this case the Cartesian product) but for which we make the following identifications:</div><div><ul><li>\((v_1, w)+(v_2, w) = (v_1 + v_2, w)\)</li><li>\((v, w_1)+(v, w_2) = (v, w_1 + w_2)\)</li><li>\(c(v,w) = (cv, w) = (v, cw)\)</li></ul><div>For physical justification think of V and W as one dimensional vector spaces corresponding to distinct dynamical degrees of freedom. Linearity is a property of vector spaces and we expect this property to be preserved if vector spaces are to describe nature. Bilinearity in the equivalence relationship above arises because the degrees of freedom are independent.</div></div><div><br /></div><div>Now a Cartesian product of vector spaces respecting the above relationships is a new mathematical object: a tensor product.</div><div><br /></div><div>The tensor product is unique up to isomorphism and respects the following <a href="https://en.wikipedia.org/wiki/Universal_property" target="_blank">universal property</a>:</div><div><br /></div><div>There is a bilinear map \(\phi : V\times W \rightarrow V\otimes W\) such that given <b>any</b> other vector space Z and a bilinear map \(h: V\times W \rightarrow Z\) there is a unique linear map \(h^{'}: V\otimes W \rightarrow Z\) such that the diagram below commutes.</div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-5hwrfRw5nIM/WM8KagnQ8wI/AAAAAAAABKo/iWiQyIfEqqU3GXb9epQIqqooEqTokZKuwCLcB/s1600/tensor.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-5hwrfRw5nIM/WM8KagnQ8wI/AAAAAAAABKo/iWiQyIfEqqU3GXb9epQIqqooEqTokZKuwCLcB/s1600/tensor.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">This universal property is very strong and several mathematical facts follows from it: the tensor product is unique up to isomorphism (instead of Z consider another tensor product \(V\otimes^{'}W\) ), the tensor product is associative, and there is a natural isomorphism between \(V\otimes W\) and \(W\otimes V\) making the tensor product an example of a <a href="https://en.wikipedia.org/wiki/Symmetric_monoidal_category" target="_blank">symmetric monoidal category</a>, just like the category of physical systems under composition.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This may look like an insignificant trivial observation, but it is extremely powerful and it is the starting point of quantum mechanics reconstruction. <b>On one hand we have composition of physical systems and theories of nature describing physical systems. On the other hand we have dynamical degrees of freedom and the rules of quantum mechanics. The two things are actually identical and each one can be derived from the other. </b>To do this we need one additional ingredient: time viewed as a functor. Please stay tuned.</div><div><br /></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-78499928397017430542017-03-13T00:21:00.000-04:002017-03-13T00:21:36.966-04:00<h2 style="text-align: center;">Category Theory Product</h2><div><br /></div><div>Before we discuss this week's topic, I want to make two remarks from the prior posts content. First, why we need natural transformations in algebraic topology? Associating groups to topological spaces (which incidentally describe the hole structure of the space) is done by the use of functors. Different (co)homology theories are basically different functors, and their equivalence is the same as proving the existence of a natural transformation. Second, the logic used in category theory is intuitionistic logic where truth is proved constructively. Since this is mapped into computer science by the Curry-Howard isomorphism, the fact that some statements have no constructive proof is equivalent with a computation running forever. In computation theory one encounters <a href="https://en.wikipedia.org/wiki/Halting_problem" target="_blank">the halting problem</a>. If the halting problem were decidable then category theory would have been mapped to ordinary logic instead of intuitionistic logic.</div><div><br /></div><div>Now back to the topic of the day. We are still in pure math domain and we are looking at mathematical objects from 10,000 feet disregarding their nature and observing only their existence and their relationships (objects and arrows). The first question one asks is how to construct new categories from existing ones? One way is to simply reverse the direction of all arrows and the resulting category is unsurprisingly called the opposite category (or the dual). Another way is to combine two category into a new one. Enter the concept of a product of two categories: \(\mathbf{C}\times \mathbf{D}\). In set theory this would correspond with to the Cartesian product of two sets. However we need to give a definition which is independent of the nature of the elements. Moreover we want to give it in a way which guarantees uniqueness up to isomorphism. </div><div><br /></div><div>The basic idea is that of a projection from the elements of \(\mathbf{C}\times \mathbf{D}\) back to the elements of \(\mathbf{C}\) and \(\mathbf{D}\). So how do we know that those projections and the product is unique up to isomorphism? Suppose that there is another category \(\mathbf{Y}\) with maps \(f_C\) and \(f_D\). Then there is a unique map \(f\) such that the diagram below commutes</div><div><br /></div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-RNLjwZadnjs/WMYT7wKhrFI/AAAAAAAABJE/IhaFCcX3po4pGii2Nm0iFq8cBfKcIt3VwCLcB/s1600/Prod.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="120" src="https://1.bp.blogspot.com/-RNLjwZadnjs/WMYT7wKhrFI/AAAAAAAABJE/IhaFCcX3po4pGii2Nm0iFq8cBfKcIt3VwCLcB/s320/Prod.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">This diagram has to commute for all categories \(\mathbf{Y}\) and their maps \(f_C\) and \(f_D\). From this definition, can you prove uniqueness of the product up to isomorphism? It is a simple matter of "diagram reasoning". Just pretend that Y is now the "true incarnation" of the product. You need to find a morphisms f from Y to CxD and a morphism g from CxD to Y such that \(g\circ f =1_{C\times D}\), \(g\circ f = 1_Y\). See? Category theory is really easy and not harder than linear algebra.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Now what happens if we flip all arrows in the diagram above? We obtain a coproduct category \(\mathbf{C}\oplus \mathbf{D}\) and the projections maps become injection maps. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">OK, time for concrete examples:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>sets: product = Cartesian product, coproduct = disjoint union</li><li>partial order sets: product = greatest lower bounds (meets), coproduct = least upper bounds (joins)</li></ul><div>So where are we now? The concept of the product is very simple, but we need it as a stepping stone to the concept of tensor product and (symmetric) monoidal category. Why? Because <b>physical systems form a symmetric monoidal category</b>. Using categorical arguments we can derive the complete mathematical properties of any theory of nature describing a symmetric monoidal category. And the answer will turn out to be: quantum mechanics. Please stay tuned.</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-87634074684798928382017-03-04T23:08:00.000-05:002017-03-05T00:44:53.316-05:00<h2 style="text-align: center;">The Curry–Howard isomorphism</h2><div><br /></div><div>Category theory may seem vary abstract and intimidating, but in fact it is extremely easy to understand. In category theory we look at concrete objects from far away without any regard for the internal structure. This is similar with Bohr's position on physics: physics is about what we can say about nature, and not decide what nature is. Surprisingly, a lot of information about the objects in category theory is derivable from the behavior of the objects and this is where I am ultimately heading with this series on category theory.</div><div><br /></div><div>Last time I mentioned the origin of category theory as the formalism to clarify when two homology theories are equivalent. But category theory can be started from two other directions as well, and those alternative viewpoints help provide the intuition needed to navigate the abstractions of category theory. One thread of discussion starts with the idea of computability and the work of <a href="https://en.wikipedia.org/wiki/Alonzo_Church" target="_blank">Alonzo Church</a> and <a href="https://en.wikipedia.org/wiki/Alan_Turing" target="_blank">Alan Turing.</a> Turing was Church's student and each started an essential line of research: <a href="https://en.wikipedia.org/wiki/Lambda_calculus" target="_blank">lambda calculus</a> and <a href="https://en.wikipedia.org/wiki/Universal_Turing_machine" target="_blank">universal Turing machines</a>. Those later grew into one hand functional languages like Java script, and the other hand into object oriented languages like C++. What one can do with lambda calculus can be achieved with universal Turing machines, and the other way around. The essential idea of computer programming is to build complex structures out of simpler building blocks. Object oriented programming starts from the idea of packaging together actions and states. An object is a "black box" containing actions (functions performing computations) and information (the internal state of the object). Functional programming on the other hand lacks the concept of an internal state and you deal only with functions which take an input, crunch the numbers, and then produce an output. The typical example is FORTRAN: FORmula TRANslation (from higher level human understandable syntax into zeroes and ones understandable by a machine).<br /><br />The second direction one can start category theory is <a href="https://en.wikipedia.org/wiki/Intuitionistic_logic" target="_blank">intuitionistic logic</a> and the foundation of set theory. The problem of naive set theory is that one can create paradoxes like <a href="https://en.wikipedia.org/wiki/Russell's_paradox" target="_blank">Russel's paradox</a>: the set of all sets which are not members of themselves. The solution Russel proposed was <a href="https://en.wikipedia.org/wiki/Type_theory" target="_blank">type theory</a>. Types introduce structure to set theory preventing self-referential constructions. In computer programming, types are semantic rules which tell us how to understand various sequences of zeros and ones from computer memory as integers, boolean variables, etc.<br /><br />In intuitionistic logic statements are not true by simply disproving their falsehood, but they are true by providing an explicit construction. Truth must be computed and the parallel with computer programming is obvious. There is a name for this relationship, the <a href="https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence" target="_blank">Curry-Howard isomorphism</a>. <b>The mathematical formalism needed to rigorously spell out this correspondence is category theory. </b>At high level:<br /><ul><li>proofs are programs</li><li>propositions are types</li></ul><div>More important is that we can attach logical and programming meaning to category theory constructions which helps dramatically reduce the difficulty of category theory to that of elementary linear algebra. </div><div><br /></div>There are two additional key points I want to make. First category theory ignores the structure of the objects: they can be sets, topological spaces, posets, even physical systems. As such <b>uniqueness is relaxed in category theory and things are unique up to isomorphisms. </b>Second, <b>we are strengthening uniqueness by seeking </b><a href="https://en.wikipedia.org/wiki/Universal_property" style="font-weight: bold;" target="_blank">universal properties</a><b>. </b>This gives category theory its abstract flavour:<b> </b>the generalization of standard mathematical concepts in category theory involve diagrams which must commute. The typical definition is something like: "if there is an "impostor" which claims to have the same properties as the concept being defined, then there exist a so and so isomorphism such that a certain diagram commutes which guarantees that the impostor is nothing but a restatement of the same concept up to isomorphism". Next time I will talk about the first key definition we need from category theory, that of a product, and by flipping the arrows that of a coproduct. </div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-41329477995214228272017-02-28T00:16:00.000-05:002017-02-28T00:16:01.799-05:00<h2 style="text-align: center;">Objects and arrows</h2><div><br /></div><div>With one day delay, let's continue the discussion about category theory. One way to look at category theory is as a generalization of the notion of equivalence: <b>category theory = equivalence on steroids</b>. </div><div><br /></div><div>It is informative to look at the original motivation for category theory and also to look at a problem around 1900. Suppose you go back in time without knowing any modern math except group theory and you are aware of <a href="https://en.wikipedia.org/wiki/M%C3%B6bius_strip" target="_blank">Mobius strip</a>, and <a href="https://en.wikipedia.org/wiki/Klein_bottle" target="_blank">Klein bottle</a>. Your task is to try to figure out what else is possible? In other words, classify all two dimensional surfaces. Who can help you on this quest? Well, clothes are two dimensional surfaces made by tailors. How do they make them? By two operations: cutting and stitching. Knowing group theory, you realize cutting and stitching are opposite operations, and they do respect the axioms of a group. This is how <a href="https://en.wikipedia.org/wiki/Homology_(mathematics)" target="_blank">homology </a>theory actually came from: associating groups with topological spaces in order to classify them. Now fast forward to 1940s, several homology theories were known and the problem was why the groups involved in them are the same? How do we axiomatize homology theory and how do we know if two homologies are equivalent? The answer lies in the concept of <b>natural transformation</b> which requires the concept of <b>functor</b>, which in turn needs the idea of a <b>category.</b> </div><div><br /></div><div>So what is a category? A category consists of objects and morphisms (arrows) such that the morphisms can be composed. Here are some examples:</div><div><br /></div><div>-examples from math:</div><div><ul><li>sets and functions</li><li>groups and group homomorphisms</li><li>Hilbert space and operators</li><li>partial order sets and monotone functions</li><li>manifolds and cobordisms</li></ul><div>-examples from logic</div></div><div><ul><li>propositions and proofs</li></ul><div>-example from physics</div><div><ul><li>physical systems and physical processes</li></ul></div>-examples from computer science<br /><ul><li>data types and programs</li></ul><div>Now a <b>functor</b> maps a category to another category by mapping objects to objects and arrows to arrows in a way that preserves structure. This is how for example in algebraic topology we associate groups to topological spaces. </div></div><div><br /></div><div>A <b>natural transformation</b> is a arrow (morphism) between functors subject to some (natural) conditions. </div><div><br /></div><div>Apart of naturality, another key concept is <b>universality </b>which means a unique (up to an isomorphism) solution to problems of constructions. We will encounter that when we will express quantum mechanics in category formalism.</div><div><br /></div><div>Category theory reveals surprising relationships:</div><div><ul><li>Cartesian products of sets are like greater lower bounds of partial order sets</li><li>Proofs in logic are like programs in functional programming</li></ul><div>Back to quantum mechanics, unitary evolution preserves information and it should be no surprise that there quantum information can be represented in diagramatic fashion. However this is not the path I am going to take and I will make use of universality in deriving quantum mechanics from a simple principle - composition: <b>a theory T describing two physical systems A and B must described the composite system A+B as well. </b>This is very intuitive principle but in the formalism of category theory it has extremely powerful mathematical consequences, spelling out the complete internal details of the theory T. <b>Quantum mechanics comes out of this in its full detail. </b></div></div><div><br /></div><div>Please stay tuned...</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-117126206166693902017-02-19T19:54:00.000-05:002017-02-19T19:54:56.768-05:00<h2 style="text-align: center;">Monoids: the root of it all</h2><div><br /></div><div>Let's start talking about category theory. We will start from set theory and in the end try to get away from it. The first thing we need to discuss is <a href="https://en.wikipedia.org/wiki/Magma_(algebra)" target="_blank">magma</a>. Basically you have a binary operation on a set and that's all: \(M \times M\rightarrow M\). One problem with magmas is that there is no associativity. Now not all mathematical operations lacking associativity are inherently primitive. Think of Lie algebras: the operation is not associative. However there you have something else: the Jacobi identity. But a pure magma without any additional structure is a rather inert object. The other problem with magmas is the lack of a unit. Add associativity and a unital element and category theory comes alive. </div><div><b><br /></b></div><div><b>To link the discussion to physics, nature obeys the structure of a (commutative) monoid: <u>two physical systems can be composed into a larger physical system:</u></b></div><div><b>- </b>composition is the binary operation </div><div><b>- </b>associativity guarantees our ability to reason about physical systems regardless of how we split a physical system into subsystems: quantum mechanics is valid for both an electron or an atom containing an electron</div><div><b>-</b> the unital element is nothingness: composition with nothing leaves the original physical system intact.</div><div><br /></div><div>In later posts I will show how quantum mechanics is a logical consequence of the commutative monoid above. In other words, quantum mechanics is inescapable and nature is quantum all the way.</div><div><br /></div><div>Back on monoids, let's fall back on the usual example: composable functions: the image of a function is the domain of the next function. The link with programming is obvious: the output of one computation is plugged in as the input of another computation. As a side note, because of this functional programming is best explained in the language of category theory. When we talk function composition we usually write: \(f \circ g\) which means \(f(g(x))\). To jump in abstraction and eliminate the nature of the elements considerations, there is an elementary trick to help navigate complex composition chains: call \(\circ\): AFTER like this: \(f~composed~with~g = f\circ g = f~ AFTER~ g\)</div><div><br /></div><div>Now let's review the usual properties of injectivity and surjectivity:</div><div><br /></div><div>Injectivity: for any elements \(x, x^{'}\), a function is injective if \(f(x) = f(x^{'}) ~implies~ x=x^{'}\)</div><div>Surjectivity: for any \(y\) in the range, there is an \(x\) in the domain such that \(f(x)=y\)</div><div><br /></div><div>So how can we abstract this away and eliminate the talk about the elements? The corresponding category theory concepts are monic and epic:</div><div><br /></div><div>Monic: a morphism is monic if for any \(g, h\) \(f\circ g = f\circ h ~implies~g=h\)</div><div>Epic: a morphism is epic if for any \(g, h\) \(g\circ f = h\circ f ~implies~g=h\)</div><div><br /></div><div>Can you prove that if \(f:X\rightarrow Y\) then \(f\) is injective if and only if it is monic and it surjective if and only if it is epic? The proof can be found in many places but it is instructive to try to prove it yourself without looking it up first as this will help you better understand category theory. </div><div><br /></div><div>The last point I want to make today is that in category theory we move away from functions into abstract morphisms. <b>The key point of morphisms is that they preserve mathematical structures.</b> As such they can be used to jump between categories of very different nature. This is how category theory is a unifying structure of mathematics where the same patterns of reasoning can be replicated from logic to computer science, to algebraic topology, to quantum mechanics.</div><div><br /></div><div>To be continued...</div><div><br /></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-28736351905012547962017-02-12T21:13:00.000-05:002017-02-12T22:13:13.150-05:00<h2 style="text-align: center;">A new way to look at mathematics</h2><div><br /></div><div>I want to start today a series of posts about <a href="https://en.wikipedia.org/wiki/Category_theory" target="_blank">category theory</a>. This is a vast area of mathematics which unifies logic, computer programming, combinatorics, cohomology, etc, and <b>quantum mechanics </b>into a cohesive paradigm. It also settles the problem of interpretation for quantum mechanics. By its very construction category theory has no need for any realism baggage. The entire mathematics can be expressed not in the language of sets (which are abstractions based on our classical intuition) but <b>in the language of categories free of any considerations about the nature of elements</b>. Regarding physics, the paradigm of category theory is best expressed by a famous Bohr quote:</div><br /><i>"It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we say about Nature."</i><br /><div><i><br /></i></div><div>Let me start slow. The usual usage of math is on the practical side to solve problems. How many times did we hear the lazy student complaint: why should we learn this? Math is not about memorization and math is very easy once we absorb its content. Learning math is a journey in mastering abstractions and general ways of reasoning. For example when you learn about Lie groups you can extract a lot of key result by elementary methods simply by studying matrices. However you hit a wall with octonions because they are not associative and do not admit a matrix representation for this very reason. In turn this precludes the proper understanding of exceptional Lie groups.</div><div><br /></div><div>Or consider a simpler example, topology. A lot of functional analysis can be done using the concept of distance and metric spaces. For example a space in \(R^n\) is compact iff it is closed and bounded. Then the metric spaces are generalized by the concept of <a href="https://en.wikipedia.org/wiki/Topological_space" target="_blank">topological space</a>s which are based on the idea of neighborhoods, unions, and intersections. In this case compactness is defined much more abstractly: a space is compact iff any open covering has a finite subcover. </div><div><br /></div><div>A similar thing happens in category theory. Patterns of reasoning in various mathematical domains are abstracted away in a formalism which does not care about the nature of the elements. On one end this is harder and to help navigate this in the beginning you hold on particular examples; the typical examples are functions. However at some point you let go of the examples just like in topology you let go the notion of distance. At that point you learn to reason properly in category theory and a lot can be achieved in this way. Then we can make the journey backwards from abstract to concrete. There is a big bonus in this: we have the flexibility to pick the concrete examples we want. And in our case we will pick quantum mechanics. <b>Quantum mechanics is best and most naturally expressed in the language of category theory. </b>Goodbye sets, goodbye classical realism, let the category journey begin. Please stay tuned. </div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-71473466697640513722017-02-04T22:00:00.000-05:002017-02-04T22:43:01.570-05:00<h2 style="text-align: center;">Trump, DeVos and the fleecing of America</h2><div><br /></div><div>Once upon a time there was a Sputnik circling the Earth and the <a href="https://en.wikipedia.org/wiki/Sputnik_crisis" target="_blank">fear </a>it created spurred America to wake up and invest massively into education. Those days are long gone and now religious extremists (like late Jerry Falwell and his son Jerry Falwell Jr) wage war on science in United States. Sadly they are about to destroy the education system and the end result will be an <a href="https://en.wikipedia.org/wiki/Idiocracy" target="_blank">Idiocracy </a>society where we study only creationism and we water plans with sport drinks because they have electrolytes - what plants crave.</div><div><br /></div><div>So what is president Trump's policy? <b>He has only one policy: to continuously demonstrate he has the largest dick.</b> </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-RA2xjHuCz54/WJZWV5yiX_I/AAAAAAAABIo/IINHx_gWz6kZ5CwFnaNY7QYbSu7CMlGZwCLcB/s1600/camacho.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="154" src="https://1.bp.blogspot.com/-RA2xjHuCz54/WJZWV5yiX_I/AAAAAAAABIo/IINHx_gWz6kZ5CwFnaNY7QYbSu7CMlGZwCLcB/s320/camacho.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">From inauguration crowd size to popular vote size, it is all about how he is "yuuuge". Help him masturbate his ego in public and you get away with anything. One such person is Betsy DeVos. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="https://en.wikipedia.org/wiki/Betsy_DeVos" target="_blank">Betsy DeVos</a> is a billionaire who made her money with <a href="https://en.wikipedia.org/wiki/Amway" target="_blank">Amway</a> and she bought her way into the Trump administration by donating to the republican party something to the tune of 200 million dollars. <b>On the recent confirmation hearings she said she is supports guns in schools to protect students from grizzly bears!!! </b>Also she did not understand the difference between the value of something and the rate of increase of that value. Her intellectual level is that of a moronic imbecile who repeatedly failed to complete 3rd grade. Honestly, she deserves a prize for managing to beat Sarah Palin in stupidity: a really really hard thing to achieve.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">So why does she want to lead the education department? Because <b>the education budget is over 140 billion dollars. Cha-ching! By refusing to hold both public and private schools accountable <u>to the same standards</u> she opens to door to scams like her boss' "Trump University" And it is all paid for by us, the taxpayers. </b>What? Did you believe those 200 million dollars were donated out of the goodness of her hearth?</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Now maybe Amway is a legit business and she is not a nutcase. Do you know how Amway works? It is a pyramid scheme going under the name "<a href="https://en.wikipedia.org/wiki/Multi-level_marketing" target="_blank">multi-level marketing</a>": you buy their soap and you sell it to say 5 of your friends and if you sign them up as Amway distributors you get a cut from their sale as well. Now you don't get rich by selling 3 dollars worth of soap in a month, but by signing up many more people and they in turn do the same. The end result is a pyramid of losers. Most of them end up broke and with a garage full of soap inventory. They loose money on books and brainwashing cult-like company seminars. So why is this not illegal? Because the Federal Trade Commission guys are crooks too: since there is an actual product flowing (which keeps the scam going) they merely get their cut of the scam by huge fines (which would have been impossible from traditional Ponzi schemes because in that case when the scheme crashes the money stops flowing). Now Amway is not alone in MLM. Here is a nice video about the dangers of multi-level marketing:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div style="text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/s6MwGeOm8iI/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/s6MwGeOm8iI?feature=player_embedded" width="320"></iframe></div><br /><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Back to DeVos. She is promoted by Trump to buy republican support for keeping him in power, but she is a pure republican creation. <b>Both republicans and Trump are willing to fleece America, their only difference is how they sell it to their base:</b> republicans trick them by appealing to freedom, self-reliance, and independence, while Trump promises to make their dicks great again. <b>Trump's ideology is nothing but the fascist utopia: "you are the best, screw everyone else" </b>(like mexicans, muslims, and recently australians)<b>.</b></div><div class="separator" style="clear: both; text-align: left;"><b><br /></b></div><div class="separator" style="clear: both; text-align: left;">I don't want to end on a dark note so here is a nice video: </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div style="text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/9Sq-VPDtNK4/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/9Sq-VPDtNK4?feature=player_embedded" width="320"></iframe></div><div style="text-align: center;"><br /></div><div style="text-align: left;">because the best way to deal with Trump is by making fun of him. There are many more videos like this for various countries in the world. Have fun watching them all.</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-52145621259178352032017-01-29T22:50:00.000-05:002017-01-29T22:50:51.501-05:00<h2 style="text-align: center;">Surreal Trajectories: the main argument against Bohmian ontology</h2><div><br /></div><div>I was extremely busy for the past two weeks and I simply did not have any time to write the weekly post. But without any more delays, as promised, here is the argument against Bohmian interpretation. The argument comes from a famous paper by Englert, Scully, Süssmann, and Walther: <i>Surrealistic Bohm Trajectories</i>.</div><div><br /></div><div>For clarity, here are the original <a href="https://www.degruyter.com/downloadpdf/j/zna.1992.47.issue-12/zna-1992-1201/zna-1992-1201.xml" target="_blank">paper</a>, the <a href="https://www.degruyter.com/downloadpdf/j/zna.1993.48.issue-12/zna-1993-1219/zna-1993-1219.xml" target="_blank">rebuttal</a>, and the <a href="https://www.degruyter.com/downloadpdf/j/zna.1993.48.issue-12/zna-1993-1220/zna-1993-1220.xml" target="_blank">response</a>.</div><div><br /></div><div>The argument is simple: in a double slit experiment with a which way detector present before the slit (which incidentally kills the interference pattern), the Bohmian trajectories do not cross the axis of symmetry. </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-qtSXrVmbuDE/WI6vy6Vis4I/AAAAAAAABIM/w4O6dRpqHq8R8PHnF8BRAFfXE2VpFRtkACLcB/s1600/DoubleSlit.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="296" src="https://1.bp.blogspot.com/-qtSXrVmbuDE/WI6vy6Vis4I/AAAAAAAABIM/w4O6dRpqHq8R8PHnF8BRAFfXE2VpFRtkACLcB/s320/DoubleSlit.png" width="320" /></a></div><div><br /></div><div><br /></div><div>However the wavefunction is:</div><div><br /></div><div>\(\Psi = \psi_>|detect~up\rangle + \psi_<|detect~down\rangle\)</div><div><br /></div><div>and \(\psi_>\) does not vanish in the bottom part and \(\psi_<\) does not vanish in the top part. <b>As such, the particle can be found in the down section while the particle was detected earlier by the upper which way detector. <u>But this is at odds with Bohmian trajectories which by symmetry considerations do not connect the up with the down.</u></b></div><div><b><u><br /></u></b></div><div>The conclusion is that Bohmian trajectories do not <i>always</i> have a correspondence in reality. The issue is not whether Bohmian quantum mechanics does or does not make the same prediction as standard quantum mechanics as the rebuttal seems to imply, but <b>the issue is the ontology of Bohmian trajectories</b>. The claimed advantage of Bohmian mechanics is its clarity rooted in realism, but <b>if Bohmian trajectories are at odds with experiments, what is the value of Bohmian interpretation?</b> <b>Remember that in Bohmian interpretation the only thing "real" is the particle trajectory</b>. I could not find a valid answer from the Bohmian community to the surrealistic paper challenge and <b><u>in my opinion this paper it is a decisive clear cut argument against Bohmian interpretation.</u></b> </div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com7tag:blogger.com,1999:blog-3832136017893749497.post-19638246834488980892017-01-16T01:04:00.000-05:002017-01-17T21:54:04.351-05:00<h2 style="text-align: center;">Book Review: "Making Sense of Quantum Mechanics"</h2><div><br /></div><div>After much delay I had found the time to finish reading Jean Bricmont's "Making Sense of Quantum Mechanics" <a href="http://www.springer.com/us/book/9783319258874" target="_blank">book</a>.</div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-u-tgekEgpoo/WHw00O_XgpI/AAAAAAAABHs/g5mVi8MWfcEONS1QK3MCgY9BhZlvMNuDACLcB/s1600/MakingSense.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-u-tgekEgpoo/WHw00O_XgpI/AAAAAAAABHs/g5mVi8MWfcEONS1QK3MCgY9BhZlvMNuDACLcB/s320/MakingSense.jpg" width="212" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div><b>The book is the best presentation of Bohmian interpretation</b> I have ever read. It masterly combines the philosophical ideas with a bit of math, famous quotes, and some historical perspective. </div><div><br /></div><div>After preliminary topics in chapter one, chapter two discusses the first quantum "mystery": superposition, while chapter four discuses the second "mystery": nonlocality. It was chapter three, a philosophical "intermetzzo" which took me a very loooong time to read and prevented me to write this review much sooner: one one end I could not write this post without reading it, and on the other end I was loosing interest very quickly in it after a couple of pages of historical review. Then Bricmont proceeds into presenting Bohniam mechanics - the heart of the book. </div><div><br /></div><div>Let's dig a bit deeper into it. Chapter two is a very well written introduction into why quantum mechanics is counter-intuitive. This is presented in the style of modern quantum foundation undergrad classes. Chapter four main idea is this: to many physicists Bell's result proved the impossibility of non-contextual hidden variables (or local realism) while Bell should be understood in conjunction with EPR: EPR+Bell = nonlocality. But what does nonlocality mean? Is it just higher than expected correlations? Here Bricmont makes a very bold and provocative claim: </div><div><br /></div><div>"<i>the conclusion of his </i>[Bell's]<i> argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between events) in Nature</i>."</div><div><br /></div><div>To support this chapter 4.2 discusses "Einstein's boxes" [I had a series of posts discussing <u>why in my opinion they do not</u> represent an argument for nonlocality. What EPR+Bell shows is that the composition of two physical systems into a larger physical system does not respect the rules of classical physics - parabolic composability but new rules - elliptic composability. Nature is not "nonlocal" but "non-parabolic composable"]. </div><div><br /></div><div>Onto the main topic, the presentation of Bohmian mechanics is standard and what it is surprising is the degree on which the underdetermination issue is addressed: there are an infinite number of alternative theories (like Nelson's stochastic theory) which are in the same realistic vein and which make the same predictions as Bohmian mechanics. Chapter three discussion is invoked here but I feel the argument is very weak (not even a handwaving).</div><div><br /></div><div>Then the book talks about alternative approaches to Bohmian mechanics courageously taking (some well deserved some not) shots at alternative interpretations (like GRW, MWI, CH, Qbism), and wraps up with historical topics and sociological arguments.</div><div><b><br /></b></div><div><b>Now onto what the book covers poorly: surreal trajectories, and quantum field theory in Bohmian mechanics. </b>Surreal trajectories are mentioned in passing in a quote while they are <i><u>the</u> </i>major objection to the interpretation. As I said before, the very name "<b>surreal trajectory</b>" was a masterful catchy clever title for a paper but it backfired in the long term because it was attaching a stigma to Bohmian mechanics which in turned allowed Bohmian supporters to summarily and unfairly dismiss the argument. I will revisit the argument in next post. <b>The key point of surreal trajectories paper is that the particle is detected where Bohmian mechanics predicts it must not go, and since the only thing "real" in Bohmian mechanics is the position of the particle, it represents a fatal blow to the Bohmian ontology.</b> Currently, to my knowledge, there is no consensus inside the Bohmian community on the proper answer the surreal trajectory paper: some deny it is a problem at all while others acknowledge the problem and propose (faulty) ideas on how to deal with it. This is similar with the situation inside the MWI camp where the big pink elephant in the room there is the notion of probabilities: some in MWI disagree it is an issue while others attempt to solve it (but fail). </div><div><br /></div><div>Quantum field theory in Bohmian mechanics is another sore point which is not properly discussed. <b>My take on the topic is that a Bohmian quantum field theory is impossible to be constructed, and I want to be proven wrong by a consistent proposal: show me the money, show me the archive paper where the problem is comprehensibly solved.</b></div><div><b><br /></b></div><div>Bad points aside, overall I liked the book, I find it stimulating, and I enjoyed very much reading it (except chapter 3 which invariably succeeded putting me to sleep). <b>The book is a must read for any person seriously interested in the foundations of quantum mechanics.</b></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com6tag:blogger.com,1999:blog-3832136017893749497.post-29365863491256703542017-01-09T00:01:00.000-05:002017-01-09T00:04:32.573-05:00<h2 style="text-align: center;">(The nonsense of) Joy Christian Reloaded</h2><div><br /></div><div>I was preparing the first physics posts of the year when I got some comments and a question on Joy Christian on an old blog <a href="http://fmoldove.blogspot.com/2015/07/joy-christians-program-of-achieving.html" target="_blank">post</a>. In the words of late <a href="https://en.wikipedia.org/wiki/Yogi_Berra" target="_blank">Yogi Berra</a>, this is "deja vu all over again". Probably the best description of Joy Christian is given by the Monty Python: The Dead Parrot sketch:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/4vuW6tQ0218/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/4vuW6tQ0218?feature=player_embedded" width="320"></iframe></div><div><br /></div><div>The question I got is the following:</div><div><br /></div><i>"I would like to understand whether the equations (67) - (75) in Joy Christian’s paper “Local Causality in a Friedmann-Robertson-Walker Spacetime” make any sense at all. I don't understand how the mathematical limes operation are carried out."</i><br /><div><i><br /></i></div><div>The paper which got past the referees by trickery is on the archive: <a href="https://arxiv.org/pdf/1405.2355v7.pdf" target="_blank">https://arxiv.org/pdf/1405.2355v7.pdf </a>and there you see the full derivation of the main faulty claim. Minus some obfuscation techniques, Eqs. 67-75 are nothing but the one-pager Joy preprint: <a href="https://arxiv.org/pdf/1103.1879v1.pdf" target="_blank">https://arxiv.org/pdf/1103.1879v1.pdf</a></div><div><br /></div><div><b>The main hand-waving trick in the "derivation" is a conversion inside of a sum of </b>\(\lambda^k\) <b>from a variable into an index</b> which amounts to adding apples to oranges and obtaining the incorrect result (see the bottom of page 8 on my preprint: <a href="https://arxiv.org/pdf/1109.0535v3.pdf" target="_blank">https://arxiv.org/pdf/1109.0535v3.pdf</a>). </div><div><br /></div><div>The mistake happens on the transition from Eq 73 to Eq 74 because the L's belong to two distinct kinds of algebras: let's call them apples and oranges. Ignoring the axb, the troubled sum term is something like this:</div><div><br /></div><div>\(L(\lambda^1) + L(\lambda^2)+L(\lambda^3)+L(\lambda^4)+L(\lambda^5)+...=\)</div><div>apple_1 + apple_2 + orange_3 +apple_4 + orange_5+...</div><div><br /></div><div>with \(\lambda^1 = +1, \lambda^2\ = +1, \lambda^3 = -1, \lambda^4 = +1, \lambda^5 = -1...\)</div><div><br /></div><div>and with the transformation rule: "apple = - oranges" when we convert to objects of the same kind (let's pick apples) we get:</div><div><br /></div><div>\(apple(\lambda^1) + apple(\lambda^2)-apple(\lambda^3)+apple(\lambda^4)-apple(\lambda^5)+...=\)</div><div>\(apple(+1) + apple(+1)-apple(-1)+apple(+1)-apple(-1)+...=\)</div><div>\(apple+ apple+apple+apple+apple+...=\)</div><div><br /></div><div><b>which no longer vanishes.</b></div><div><b><br /></b></div><div><b>The preparation for this trick is on Eq. 49 which encodes the two <u>distinct</u> algebras (of apples and oranges) into a common formula. </b>In my preprint you can double check this by trying out the matrix representations of the two algebras (eqs 53-56).</div><div><br /></div><div>Hopefully my explanation is clear enough. I know all Joy's mathematical tricks in all of his papers or in his blog debates, but I ran out of energy debunking his nonsense. <b>Kudos to Richard Gill for pursuing this further. </b>I was aware of the "<i>Causality in a Friedmann-Robertson-Walker Spacetime" </i>paper and it was on my to do list to write a rebuttal to it, but the journal withdrew it before I could get to it.</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com2tag:blogger.com,1999:blog-3832136017893749497.post-43419454718540524142017-01-01T18:00:00.000-05:002017-01-01T18:00:13.774-05:00<h2 style="text-align: center;">Happy New Year!</h2><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-7q1TIY8lkfE/WGmJ1aI8FGI/AAAAAAAABHU/fWxPWIIUkrcRIVQCmXgc5NCp5lWIBnS-QCLcB/s1600/2017.jpe" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-7q1TIY8lkfE/WGmJ1aI8FGI/AAAAAAAABHU/fWxPWIIUkrcRIVQCmXgc5NCp5lWIBnS-QCLcB/s320/2017.jpe" width="320" /></a></div><div><br /></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-54024425391928247202016-12-18T23:01:00.000-05:002016-12-18T23:01:46.463-05:00<h2 style="text-align: center;">Is the Decoherent Histories Approach Consistent?</h2><div><br /></div><div>One particular approach of interpreting quantum mechanics is Decoherent Histories. All major non-Copenhagen approaches have serious issues: </div><div>- MWI has the issue of the very meaning of probability and without a non-circular derivation (impossible in my opinion) of Born rule does not qualify for anything but a "work in progress" status.</div><div>-GRW-type theories make different predictions than quantum mechanics which soon will be confirmed or rejected by ongoing experiments. (My bet is on rejection since later GRW versions tuned their free parameters to avoid collision with known experimental facts instead of making a falsifiable prediction) </div><div>-Bohmian approach has issues with "surreal trajectories" which invalidates their only hard ontic claim: the position of the particle.</div><div><br /></div><div>Now onto Decoherent Histories. <b>I did not closely follow this approach and I cannot state for sure if there are genuine issues here</b>, but I can present the debate. On one hand, Robert Griffiths states: </div><div><br /></div><div>"<i>What is different is that by employing suitable families of histories one can show that measurement actually measure something that is there, rather than producing a mysterious collapse of the wave function</i>"</div><div><br /></div><div>On the other hand he states:</div><div><br /></div><div>"<i>Any description of the properties of an isolated physical system must consists of propositions belonging together to a common consistent logic</i>" - in other words he introduces contextuality.</div><div><b><br /></b></div><div><b>Critics of decoherent (or consistent) histories use examples which are locally consistent but globally inconsistent to criticize the interpretation.</b></div><div><b><br /></b></div><div>Here is an example by Goldstein (other examples are known). The example can be found in Bricmont's recent book: <a href="https://www.amazon.com/Making-Sense-Quantum-Mechanics-Bricmont/dp/3319258877" target="_blank">Making Sense of Quantum Mechanics</a> on page 231. Consider two particles and two basis for a two-dimensional spin base \((|e_1\rangle\, |e_2\rangle), (|f_1\rangle, |f_2\rangle))\) and consider the following state:</div><div><br /></div><div>\(|\Psi\rangle = a |e_1\rangle|f_2\rangle + a|e_2\rangle|f_1\rangle - b |e_1\rangle|f_1\rangle\)</div><div><br /></div><div>Then consider four measurements A, B, C, D corresponding to projectors on four vectors, respectively: \(|h\rangle, |g\rangle, |e_2\rangle, |f_2\rangle\) where:</div><div><br /></div><div>\(|g\rangle = c|e_1\rangle + d|e_2\rangle\)</div><div><div>\(|h\rangle = c|f_1\rangle + d|f_2\rangle\)</div></div><div><br /></div><div>Then we have the following properties:</div><div><br /></div><div>(1) A and C can be measured simultaneously, and if A=1 then C=1 </div><div><div>(2) B and D can be measured simultaneously, and if B=1 then D=1</div><div>(3) C and D can be measured simultaneously, but we never get both C and D = 1</div><div>(4) A and B can be measured simultaneously, and sometimes we get both A and B = 1</div><div><br /></div><div><b>However all 4 statements cannot be true at the same time: </b>when A=B=1 as in (4) then by (1) and (2) C=D=1 and this contradicts (3).</div><div><br /></div><div>So what is going on here? The mathematical formalism of decoherent histories is correct as they predict nothing different than standard quantum mechanics. <b>The interpretation assigns probabilities to events weather we observe them or not, but does it only after taking into account contextuality. </b>Is this a mortal sin of the approach? Nature is contextual and I don't get the point of the criticism. The interpretation would be incorrect if it does not take into account contextuality. Again, <b>I am not an expert of this approach and I cannot offer a definite conclusion, but to state my bias I like the approach and my gut feeling is that the criticism is without merit.</b></div><div> </div></div><div><span style="font-size: x-small;">PS: I'll be going on vacation soon and my next post will be delayed: I will skip a week.</span></div><div><br /></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com4tag:blogger.com,1999:blog-3832136017893749497.post-22287855486165695582016-12-11T16:09:00.000-05:002016-12-11T21:39:53.006-05:00<h2 style="text-align: center;">Measurement = collapse + irreversibility</h2><div><br /></div><div>I got a lot of feedback from last 2 posts and I need to continue the discussion. Even Lubos with his closed mind unable to comprehend anything different than the textbooks from 50 years ago and his combative style said something worth discussing.</div><div><br /></div><div>But first let me thank Cristi for the picture below which will help clarify what I am trying to state. Let me quickly explain it: the interferometer arms are like the two sides of Einstein's box and once the particle was launched -for the duration of the flight- you can close the input and output of the interferometer, open the exit just in time and still have the interference. So this seems to contradict my prediction. But does it?</div><div><br /></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-M96UtS6YIy0/WE2MmpG9kdI/AAAAAAAABGw/EJvYfa1vI2Q_4McRSCryJgU3uHiHSiaKQCLcB/s1600/K4zVMhj.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="174" src="https://4.bp.blogspot.com/-M96UtS6YIy0/WE2MmpG9kdI/AAAAAAAABGw/EJvYfa1vI2Q_4McRSCryJgU3uHiHSiaKQCLcB/s400/K4zVMhj.png" width="400" /></a></div><div><br /></div><div><br /></div><div>This time I do need to dig a bit deeper into the mathematical formalism. First, the role of the observer is paramount: <b>no observer</b>=<b>no measurement. </b>Second, the observer is described by quantum mechanics as well: there is the wavefunction of the quantum system, and there is the wavefunction of the observer. Now here is the new part: while we can combine the quantum system and the observer by tensor product and do the usual discussion of how unitary evolution does not predict an unique outcome, <b>we need to combine the quantum system and the observer using the Cartesian product</b>. This is something new, not present in standard quantum mechanics textbooks. However this follows naturally from the category theory derivation of quantum mechanics from first principles. There are equivalent Cartesian products corresponding to potential measurement outcomes:</div><div><br /></div><div>\((|collapsed ~1 \rangle, | observer~ see~1\rangle ) \equiv ( |collapsed~2\rangle, | observer~see~2\rangle)\) </div><div><br /></div><div>This equivalence exists in a precise mathematical sense and respects the three properties of the equivalence relationship: reflexivity, symmetry, and transitivity. <u style="font-weight: bold;">Break the Cartesian pair equivalence by any mechanism possible and you get the collapse of the wavefunction.</u> </div><div><br /></div><div><b>Closing the interferometer, or cutting Einstein's box in half kills the equivalence and the wavefunction collapses. </b>However while the particle is still in flight the process is reversible!!! Open the interferometer's exits in time and you restore the equivalence, undo the collapse still get the interference (Han Solo kills the stormtrooper 100% of the time). </div><div><br /></div><div>However there is a way to make the collapse permanent: just wait enough time with the ends closed such that the energy-time uncertainty relation allows you to reduce the energy uncertainty to the point that you can detect the particle inside by weighing the half-boxes or the arms of the interferometer. Suppose the ends of the interferometer is made out of perfect mirrors.<b> <u>If you wait long enough (even though you are not physically weighing anything) and then reopen the exits will result in loss of interference: this is my prediction.</u></b></div><div><b><br /></b></div><div>But what happens if you only wait a little bit of time and you are in between full interference and no interference? You get a <a href="https://en.wikipedia.org/wiki/Weak_measurement" target="_blank">weak measurement</a>.</div><div><br /></div><div>Now let me discuss Lubos objection and then come back to the nonlocality point Bricmont was making.<br /><br />First Lubos was stating: "If you place some objects (a wall) at places where a particle is certain not to be located, the effect on the particle's future behavior is obviously non-existent". The objection is vacuous. Obviously I don't disagree with the statement, but his entire line of argument is incorrect because <b>collapse is not a dynamic process. If collapse would have had a dynamic origin then we would have had a unitary explanation for it we would have had to talk about the "propagation" of collapse. </b>What the Cartesian pair mathematical framework does is first getting the rid of the consciousness factor, and second clarifying the precise mathematical framework of how the observer should be treated within the formalism. <b>Contextuality is paramount in quantum mechanics</b><b> and cutting the box changes the context of the experiment.</b><br /><b><br /></b>Now onto Bricmont argument. Andrei stated in his comments: " I still do not see the relevance of all this in regards to the locality dilemma.". It has deep relevance as I will explain. And by the way, the rest of Andrei's comments were simply not worth answering-nothing personal: I don't have the luxury of enough free time to answer each and every comment.<br /><br />Bricmont's point on Einstein's boxes was this: "either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete "<br /><br />Let's discuss the two options:<br /><ol><li>opening B1 changes the situation at B2</li><li>or the particle was in B2 all along and quantum mechanics is incomplete</li></ol><div>Option one is clearly not the case. <b>Wait long enough and the interference will no longer happen. At that point the particle IS in either B1 or B2 and shipping one box far away changes nothing.</b> But how about option 2? Is quantum mechanics incomplete? Bohmian supporters think so because they augment the wavefunction with a hidden variable: the particle's initial position. <b>Do we actually need this initial condition to make predictions? Not at all. </b>Last thing to consider: was the particle in say B2 all along? If yes, there is no interference because the which way information. What about weak measurements? This is a case of even more examples of "surrealistic trajectories": combine two interferometers and you can obtain disjoint paths!!! <b>The only thing which makes sense is that the particle does not have a well defined trajectory.</b></div><div><b><br /></b></div><div>My question to Bohmian interpretation supporters is as follows: In the above picture close the arms long enough. What happens to the quantum potential? Does it dissipate? If yes how? If no, do you <i>always</i> get interference after opening the stormtrooper end regardless of the wait time? </div><div><br /></div><div>Finally back to measurement. There is no (strong) measurement without collapse. Collapse happens when a particular equivalence relationship no longer holds. Mathematically it can be proven that the wavefunction is projected on a subspace of the original Hilbert space. Moreover uniqueness can be proven as well: that this is the only mathematically valid mechanism in which projection can occur. Interaction between the quantum system and the measurement device can break the equivalence, but changing the experimental context can achieve the same thing as well. <b>A measurement does not happen however until irreversibility occurs: there could be amplification effects, or as above enough time passes such that the energy uncertainty is low enough and the "which way" information becomes available regardless if we seek this information or not.</b></div></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com5tag:blogger.com,1999:blog-3832136017893749497.post-13472678680195663572016-12-04T21:55:00.000-05:002016-12-04T21:55:14.037-05:00<h2 style="text-align: center;">A measurement can be more than an observer learning the value of a physical observable</h2><div><br /></div><div>Last post created quite a stir and I want to expand on the ideas from it. This will also help me get out of an somewhat embarrassing situation. For months now Lubos Motl tried to get revenge on his bruised ego after a well deserved April Fool's joke and became a pest at this blog. The problem is that although I have yet to see a physics post at his blog that is 100% correct, we share roughly the same intuition about quantum mechanics: I agree more much more with his position than say with the Bohmian, GRW, or MWI approaches. The differences are on the finer points and I found his in depth knowledge rusty and outdated. For his purpose: to discredit the opposite points of view at all costs this is enough, but it does not work if you are a genuine seeker of truth.</div><div><br /></div><div>So last time he commented here: <i>"A measurement is a process when an observer actually learns the value of a physical observable" </i>which from 10,000 feet is enough. However this is not precise enough, and now I do have a fundamental disagreement with Lubos which hopefully will put enough distance between him and me. </div><div><br /></div><div>More important than my little feud with Lubos, <b>I can now propose an experiment which will either validate or reject my proposed solution to the measurement problem</b>. I do have a novel proposal on how to solve the measurement problem and this is distinct from all other approaches. I was searching for months for a case of a novel experimental prediction, but when I applied it to many problems I was getting the same predictions as standard quantum mechanics. Here is however a case where my predictions are distinct. I will not work out the math and instead let me simply present the experiment and make my experimental claim.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-1tanPi_p8Nk/WETROtWLohI/AAAAAAAABGQ/GDmxwT-4yygYFftvjijk0DN9iQkCOVpQQCLcB/s1600/experiment.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="192" src="https://1.bp.blogspot.com/-1tanPi_p8Nk/WETROtWLohI/AAAAAAAABGQ/GDmxwT-4yygYFftvjijk0DN9iQkCOVpQQCLcB/s320/experiment.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Have a box with a single particle inside. The box has a middle separator and also two slits A and B which can be placed next to a two-slit screen. We can then carry two kinds of experiments:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"></div><ol><li>open the two slits A and B without dropping the separator allowing the particle to escape the box and hit a detector screen after the two-slit screen.</li><li>drop the separator and then open the two slits A and B allowing the particle to escape the box and hit a detector screen after the two-slit screen.</li></ol><div>Next we repeat the experiments 1 or 2 enough times to see the pattern emerge on the final screen. <b>Which pattern would we observe?</b></div><div><b><br /></b></div><div>For experiment 1 we already know the answer: if we repeat it many times we obtain the interference pattern, but what will we get in the case of experiment number 2?</div><div><br /></div><div><b>If dropping the separator constitutes a measurement, the wavefunction would collapse and we get two spots on the detector screen corresponding to two single slit experiments. If however </b><b>dropping the separator does not constitute a measurement, then we would get the same interference pattern as in experiment 1.</b></div><div><b><br /></b></div><div>My prediction (distinct from textbook quantum mechanics) is that there will be no interference pattern.</div><br /><div><br /></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com21tag:blogger.com,1999:blog-3832136017893749497.post-55141866205976462882016-11-28T00:07:00.000-05:002016-11-28T00:07:52.801-05:00<h2 style="text-align: center;">Are Einstein's Boxes an argument for nonlocality?</h2><h3 style="text-align: center;">(an experimental proposal)</h3><div><br /></div><div>Today I want to discuss a topic from an excellent book by Jean Bricmont: <a href="http://www.springer.com/us/book/9783319258874" target="_blank">Making Sense of Quantum Mechanics</a> which presents the best arguments for the Bohmian interpretation. Although I do not agree with this approach I appreciate the clarity of the arguments and I want to present my counter argument.</div><div><br /></div><div>On page 112 there is the following statement: "... <i>the conclusion of his [Bell] argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between distant events) in Nature</i>". </div><div><br /></div><div>To simplify the argument to its bare essentials, a thought experiment is presented in section 4.2: Einstein's boxes. Here is how the argument goes: start with a box B and a particle in the box, then cut the box into two half-boxes B1 and B2. If the original state is \(|B\rangle\), after cutting the state it becomes:</div><div><br /></div><div>\(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) </div><div><br /></div><div>Then the two halves are spatially separated and one box is opened. Of course the expected thing happens: the particle is always found in one of the half-boxes. Now suppose we find the particle in B2. Here is the dilemma: either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete because \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) does not describe what is going on. <b>My take on this is that the dilemma is incorrect. Splitting the box amounts to a measurement regardless if you look inside the boxes or not and the particle will be in either B1 or B2. </b></div><div><b><br /></b></div><div>Here is an experimental proposal to prove that after cutting the box the state is not \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\):</div><div><br /></div><div>split the box and connect the two halves to two arms of a Mach-Zehnder interferometer (bypassing the first beam splitter). <b>Do you get interference or not? I say you will not get any interference because by weighing the boxes before releasing the particle inside the interferometer gives you the which way information.</b></div><div><b><br /></b></div><div>If we do not physically split the box, then indeed \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\), but if we do physically split it \(|B\rangle \neq \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a hidden assumption in Einstein's boxes argument: realism which demands non-contextuality. <b>Nature and quantum mechanics is contextual: w</b><b>hen we do introduce the divider the experimental context changes.</b> </div><div><br /></div><div>Bohmian's supporters will argue that always \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a simple way to convince me I am wrong: do the experiment above and show you can tune the M-Z interferometer in such a way that there is destructive interference preventing the particle to exit at one detector.</div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com13tag:blogger.com,1999:blog-3832136017893749497.post-57035089679493907032016-11-20T20:45:00.000-05:002016-11-21T11:34:40.453-05:00<h2 style="text-align: center;">Gleason's Theorem</h2><div><br /></div><div>It feels good to be back to physics, and as a side note going forward I will do the weekly posts on Sunday. Today I want to talk about <a href="https://en.wikipedia.org/wiki/Gleason's_theorem" target="_blank">Gleason's theorem</a>. But what is Gleason's theorem?</div><div><br /></div><div>If you want to assign a non-negative real valued function \(p(v)\) to every vector v of a Hilbert space H of dimension greater than two, then subject to some natural conditions the only possible choice is \(p(v) = {|\langle v|w \rangle |}^{2}\) for all vectors v and an arbitrary but fixed vector w.</div><div><br /></div><div>Therefore there is no alternative in quantum mechanics to compute the average value of an observable A the standard way by using:</div><div><br /></div><div>\(\langle A \rangle = Tr (\rho A)\)</div><div><br /></div><div>where \(\rho\) is the density matrix which depends only on the preparation process. </div><div><br /></div><div>Gleason's theorem is rather abstract and we need to unpack its physical intuition and the mathematical gist of the argument. Physically, Gleason's theorem comes from three axioms:</div><div><br /></div><div><ul><li>Projectors are interpreted as quantum propositions</li><li>Compatible experiments correspond to commuting projectors</li><li>KEY REQUIREMENT: For any two orthogonal projectors P, Q, the sum of their expectation values is the expectation value of P+Q: \(\langle P \rangle + \langle Q\rangle = \langle P+Q\rangle\)</li></ul></div><div>In an <a href="http://fmoldove.blogspot.com/2016/10/von-neumann-gleason-vs.html" target="_blank">earlier post</a> I showed how violating the last axiom (<b>which is the nontrivial one</b>), in the case of spin one particles, can be used to send signals faster than the speed of light and violate causality. But how does Gleason arrives at his result?<br /><br />Let's return at the original problem: to obtain a real non-negative function p. Now <b>add the key requirement</b> and demand that for any complete orthonormal basis \(e_m\) we have: <br /><br />\(\sum_m p(e_m) = 1\)<br /><br />For example in two dimensions on a unit circle we must have:<br /><br />\(p (\theta) + p(\theta + \pi/2) = 1\)<br /><br />which constrain the Fourier expansion of \(p (\theta)\) such that only components 2, 6, 10, etc can be non zero. <b>In three dimensions the constraints are much more severe</b> and this involves rotations under SO(3) and spherical harmonics. I'll skip the tedious math, but it is not terribly difficult to show that the only allowed spherical harmonics must be of order 0 and 2 which yields: \(p(v) = {|\langle v|w \rangle |}^{2}\).<br /><br />The real math heavy lifting is on dimensions larger than three and to prove it Gleason first generalizes \(\sum_m p(e_m) = 1\) to \(\sum_m f(e_m) = k\) where k is any positive value. He names this "f" a "frame function". Then he proceeds to show that dimensions larger than three do not add anything new.<br /><br />If you are satisfied with the Hilbert spaces of dimension 3, the proof of the theorem is not above undergrad level, and I hope it is clear what the argument is. But what about Many Worlds Interpretation? Can we use Gleason's theorem there to prove Born rule? Nope. <b>The very notion of probabilities is undefined in MWI, and I am yet to see a non-circular derivation of Born rule in MWI. </b>I contend it can't be done because it is a mathematical impossibility and I blogged about it in the past.<b> </b></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com0tag:blogger.com,1999:blog-3832136017893749497.post-28967942674350013882016-11-13T23:13:00.000-05:002016-11-13T23:17:23.112-05:00<h2 style="text-align: center;">The Open Society and Its Enemies</h2><div><br /></div><div>Today I had planned to return to physics and talk about Gleason's theorem, but as US politics still brutally interjects into our life I want to explain my take of the events and to cleanse and de-Trumpify my life before resuming the physics topics. Gleason's theorem s not going anywhere and I will postpone the topic for a week.</div><div><br /></div><div>Let me start with a "disclaimer": I am manipulated/incited by media, I am a professional protester paid by Soros, I am a hard core liberal/socialist in love with Hilary, I hate the silent majority of blue color hard working people who makes my life comfortable, and I think they are all bigots and racists. (for the record this was tongue-in-cheek).<br /><br />Until my second year in college I lived under a totalitarian system and I experienced a lengthy decade of a transition to democracy after 1989 in Romania. What I observe today in the US is the process in reverse. So how did we get here?<br /><br />To shorten the history, let me start with the end of second Bush presidency and the inauguration of Obama. At that time all inter-bank landing came to a screeching halt and it was as if someone had hit the turn-off switch on the economy. To jolt the system back to life, Obama resorted to the <a href="https://en.wikipedia.org/wiki/Keynesian_economics" target="_blank">ides of Keynes</a> and introduced the stimulus package. But <b>Obama had one fault: he was black</b> and this startled the rednecks who organized into what later became the <a href="https://en.wikipedia.org/wiki/Tea_Party_movement" target="_blank">Tea Party</a>. <b>This was America's reactionary ideology which cannot expressed their opposition to Obama's skin color due to political correctness, and instead went after him on fiscal ideas.</b> The mainstream republicans had a love/hate relationship with the Tea Party because on one hand they fear it as something which cannot be controlled, but on the other hand they draw their support from the same electoral pool.<br /><br /><b>The mainstream republicans are masters of duplicity</b>: they draw their support from the poor rural, uneducated part of America by praising their self-reliance and tickling their self-esteem, while they push policies which actually hurt their mass constituency while enriching the big business donors. To pull this trick they rely on <b>a disgusting propaganda machine: Fox News</b>. What Trump did was to break the republican lies and come out in the open with a full display of racism and intolerance. <b>Trump was running not only against Hillary but against mainstream republicans as well.</b><br /><b><br /></b>Now fast forward to today. I don't think Trump is stupid: he is a master manipulator, an immoral con man who plays on others core beliefs. <b>Trump lacks any core beliefs/moral compass and this makes him extremely dangerous: <u>a narcissistic psychopath bully now with nukes</u></b>. You only fool yourself into giving him a chance/benefit of the doubt. <b><u>The only thing Trump respects is raw power and a forceful push back.</u></b><br /><b><u><br /></u></b>But what about the republican electorate? <u style="font-weight: bold;">Almost half of them are brainwashed imbeciles:</u> in June of 2016 41% of registered republicans thought Obama was not born in US. Moreover 31% did not know what to think and only 27% of them were on the sane side!!!<br /><br />So how can we deal with Trump and his constituency? <b>Let's go back to basics, the US constitution. </b>Trump stated that the second amendment is under siege, but he is now attacking the first amendment:<br /><br /><i>"Prohibits Congress from making any law respecting an establishment of religion, impeding the free exercise of religion, <b>abridging the freedom of speech, infringing on the freedom of the press, interfering with the right to peaceably assemble</b> or prohibiting the petitioning for a governmental redress of grievances."</i></div><div><i><br /></i></div><div>The main characteristic of a totalitarian state was brilliantly captured by Popper in his book: </div><div style="text-align: left;"><a href="https://en.wikipedia.org/wiki/The_Open_Society_and_Its_Enemies" target="_blank">"The Open Society and Its Enemies"</a><br /><br />The other day Trump tweeted: "<i>Now professional protesters, incited by the media, are protesting. Very unfair!</i>"<br /><br />He tried to intimidate the media and interfered with the rights of the people to protest. Trumps wants to build physical and economic walls, deport millions of people, and wants to turn America into a closed totalitarian society. I find this unacceptable:<br /><br /><div style="text-align: center;"><b><u>Trump is not my president.</u></b></div><b><u><br /></u></b><b>Trump is not alone.</b> He has a cohort of bad supporters. First <b>the shame list of totalitarians willing to trample your rights:</b><br /><br />- Trump: see above<br />- Rudy Giuliani: advocates locking up political opponents<br />- <a href="https://en.wikipedia.org/wiki/Fort_Lee_lane_closure_scandal" target="_blank">Chris Christie</a>: did political revenge<br />- Stephan Bannon: antisemitic racist; the Goebbles of Trump<br /><br />Next this is followed by opportunistic, lying, immoral, disgusting individuals:<br />-Mike Pence: does not blink while lying to your face<br />-Reince Priebus: no slimy job is too slimy for him<br />-Bill O'Reilly<br /><br />Then plain toxic people:<br />-Sarah Palin<br />-Ted Cruz<br /><br />Then are the deplorables, and here I name a troll of this site: Lubos Motl. In Romania there is one guy Radu Moraru with his TV station Nasul TV who got involved into US politics and the Romanian vote here (in a covert effort to unseat the head of the Romanian anti-corruption agency). On that TV station I respect only one guy: Grigore Cartianu. Then there are the brainwashed.<br /><br /><div style="text-align: center;"><i>I have no respect for the people above and I draw the line here. </i></div><b><br /></b><b>The rest of people who voted for Trump are not racists or brainwashed and I have a meaningful polite conversation with them. I have several friends who voted for Trump and I have no ill feeling towards them and while we disagree we do it in the boundary of decency.</b><br /><b><u><br /></u></b>I only insist on one point: <u style="font-weight: bold;">if you voted for Trump you are personally responsible for the consequences of Trump's presidency.</u></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com15tag:blogger.com,1999:blog-3832136017893749497.post-71480413776802582582016-11-11T00:25:00.001-05:002016-11-11T00:36:11.375-05:00<h2 style="clear: both; text-align: center;">The beginning of the end of US democracy</h2><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">It's already happening...</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-kcy5nwC_QIw/WCVYrLhMHBI/AAAAAAAABFk/gnxj0mhS15Y9SI0SKY5LctaOEEcuiAfowCLcB/s1600/T1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="275" src="https://4.bp.blogspot.com/-kcy5nwC_QIw/WCVYrLhMHBI/AAAAAAAABFk/gnxj0mhS15Y9SI0SKY5LctaOEEcuiAfowCLcB/s400/T1.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Trump just visited Obama in the White House today and as a result he felt embolden to complain about protests against him. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"></div><div style="-webkit-text-stroke-width: 0px; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"></div><br /><div class="separator" style="-webkit-text-stroke-width: 0px; clear: both; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: normal; letter-spacing: normal; margin: 0px; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><u style="font-weight: bold;">If you voted for Trump you are personally responsible for the consequences of Trump's presidency.</u> </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><br />Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com1tag:blogger.com,1999:blog-3832136017893749497.post-60637732297020368822016-11-09T21:20:00.000-05:002016-11-09T21:20:24.613-05:00<h2 style="text-align: center;">Donald Trump proves Time Travel does not exist</h2><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Ypf_RN7h0eg/WCPTRfWZHeI/AAAAAAAABE8/4TJMZv3PqCIzQvr2Oc4yH2rvd4JwoPJoQCLcB/s1600/trump.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="306" src="https://1.bp.blogspot.com/-Ypf_RN7h0eg/WCPTRfWZHeI/AAAAAAAABE8/4TJMZv3PqCIzQvr2Oc4yH2rvd4JwoPJoQCLcB/s400/trump.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">I have been proven wrong, Donald Trump won the election and apparently it is not wise to underestimate people's stupidity. So how did we got here?</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">One one hand, the establishment IS corrupt and there was a deep need for fresh air. Another Bush or another Clinton was a insupportable option. But on the other hand, racism still runs deep and while it was driven underground by politically correctness, it came back with a vengeance. While Trump's disgusting brand of politics is nothing new in Europe, US did not have the antibodies to combat it and people will lean the hard way how to do it. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><u style="font-weight: bold;">If you voted for him you are personally responsible for the consequences of Trump's presidency.</u> </div><div><br /></div>Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com4tag:blogger.com,1999:blog-3832136017893749497.post-22590523356477885102016-11-08T21:23:00.004-05:002016-11-08T21:23:48.863-05:00<h2 style="text-align: center;">Waiting for US election results</h2><br />I will comment tomorrow on the election outcome...Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com7tag:blogger.com,1999:blog-3832136017893749497.post-56764169867312442692016-10-30T20:38:00.000-04:002016-10-30T20:38:34.994-04:00<h2 style="text-align: center;">Einstein's reality criterion</h2><div style="text-align: center;"><br /></div>Sorry for the delay, this week I want to have a short post continuing to showcase the results of <a href="http://tx.technion.ac.il/~peres/" target="_blank">late Asher Peres </a>which unfortunately are not well known and this is a shame.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-_SmStFLO8A8/WBaRUxtQrRI/AAAAAAAABEk/YZDa_bP8oQoOv7LG05IoeH2fh44dkzzyACLcB/s1600/quote-quantum.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="187" src="https://2.bp.blogspot.com/-_SmStFLO8A8/WBaRUxtQrRI/AAAAAAAABEk/YZDa_bP8oQoOv7LG05IoeH2fh44dkzzyACLcB/s400/quote-quantum.jpg" width="400" /></a></div><br /><br />In the famous EPR paper, Einstein introduced a reality criterion:<br /><br /><i>"If, without in any way disturbing a system, we can predict with certainty ... the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity"</i><br /><i><br /></i>Now it is generally accepted that the difference between classical and quantum mechanics is noncomutativity. While there are some subtle points to be made about this assertion (by the mathematical community), from the physics point of view the distinction is rock solid and we can build in confidence upon it.<br /><br />Now consider again the EPR-B experiment with its singlet state. Suppose the x and y components of the spin exists independent of measurement and let's call the measurement values: \(m_{1x}, m_{1y}, m_{2x}, m_{2y} \). From experimental results we know:<br /><br />\(m_{1x} = - m_{2x}\)<br />and<br />\(m_{1y} = - m_{2y}\)<br /><br />And now for the singlet state \(\psi\) let's compute:<br /><br />\((\sigma_{1x} \sigma_{2y} + \sigma_{1y} \sigma_{2x})\psi\)<br /><br />which turns out to be zero. The beauty of this is that \(\sigma_{1x} \sigma_{2y} \) commutes with \(\sigma_{1y} \sigma_{2x}\) and by Einstein's reality criterion extended to commuting operators it implies that \(m_{1x} m_{2y} = - m_{1y} m_{2x}\) <b>which contradicts</b> \(m_{1x} = - m_{2x}\) and \(m_{1y} = - m_{2y}\)<br /><br />This contradiction is in the same vein as the <a href="https://en.wikipedia.org/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state" target="_blank">GHZ</a> result, but it is not well known. The catch of this result is that measuring S1xS2y cannot be done at the same time with a measurement of S1yS2x and so <b>we are reasoning counterfactually</b>. <b>However, counterfactual reasoning is allowed in a noncontextual setting (in classical mechanics and in quantum mechanics for commuting operators) and the result is valid. </b><br /><br /><br />Florin Moldoveanuhttps://plus.google.com/117661996860849443251noreply@blogger.com12