It's about time

After discussing some well known results in quantum mechanics, I will start presenting my approach for solving the measurement problem which was my talk at the Vaxjo conference. This will take several posts, and today I start the discussion by presenting some ideas about time.

Time is essential in Hamiltonian mechanics which is the natural way to transition from classical to quantum mechanics. Usually one is introduced to the subject by the naive statement that we replace the Poisson bracket with the commutator. The story is much more subtle (and interesting) than that and today I want to explore only some parts of those issues with the remaining advanced topics to be discussed at a later time.

It turns out that there is a circularity problem: time is derived from quantum mechanics and quantum mechanics is derived from time. Let me state upfront that I do not yet know how to solve this very interesting and hard problem.

Today I'll present the claim that time has a quantum mechanical origin. The main proponents of this are Alain Connes and Carlo Rovelli in their Thermal Time Hypothesis. I have talked about this last year, but now I want to go in depth in the mathematical reasoning. As a pre-requisite the reader should understand the concept of short exact sequences and I have a review of this topic here.

I am not attempting to justify the canonical gravity point of view and oppose it to string theory, but I want to present the mathematical reasons of why time may have a non-commutative (quantum mechanical) origin. The mathematical underpinnings are von Neumann algebras and the Tomita-Takesaki theory.

The von Neumann algebras are the generalization of measure theory: they reduce to the study of measureable spaces when we restrict to the commutative case.

Suppose \(M\) is a von Neumann algebra in a Hilbert space H, \(a\) is an element of \(M\), and we have an operator S defined by:

\(S a \psi = a^* \psi\)

Then S admits a polar decomposition:

\(S = J \Delta^{1/2}\)

with \(\Delta\) a positive self-adjoined operator and J anti-unitary.

The Tomita-Takesaki theory proves that:

\(JMJ = M^{'}\): M has the same size as its commutant
\(\Delta^{-it} M \Delta^{it} = M\): there is an one-parameter group of automorphism  \(\sigma^\phi_t\) which gives the time flow in the Heisenberg picture.

So far so good, but stronger statements can be made. If we impose the KMS condition, then the automorphism is unique. Still, it depends on the choice of the state \(\phi\). The really powerful result is that in the exact sequence:

\(1\rightarrow Inn(M) \rightarrow Aut(M) \rightarrow Out(M) \rightarrow 1\)

where \(Inn(M)\) is the normal subgroup of inner automorphisms: \(T\rightarrow aTa^*\), the automorphism \(\sigma^\phi_t\) does not depend upon the choice of state \(\phi\) and hence it is a canonical time evolution.

This is the main mathematical result (by Connes) which needs to be applied to physics to understand what is going on. And this is what Connes and Rovelly do in their thermal time hypothesis paper. But there is more to the story. The short exact sequence from above was the starting point of Connes non-commutative geometry description of the Standard Model which led to the geometric unification of gauge theory weakly coupled with gravity in the non-commutative framework (no quantum gravity here: no background independence arguments to justify loop quantum gravity vs. string theory). In the simplest model of the noncommutative description of the \(SU(n)\) gauge group weakly coupled with non-quantized gravity, the short exact sequence from above is equivalent with:

\( 1 \rightarrow \mathcal{G} = Map(M, SU(n)) \rightarrow Diff(X) \rightarrow Diff(M) \rightarrow 1\)

\(1\rightarrow fiber\rightarrow total space\rightarrow base\rightarrow 1\)

where M here is a spin Riemannian manifold. The full group of invariance on a new space \( X = M \times M_n (C) \) is the semidirect product of the diffeomorphisms on M with the gauge group. The diffeomorphism shuffles (acts on) the group of gauge transformations.

If we take the main lesson of quantum mechanics to be that of non-commutativity of operators, we can construct a generalization of commutative mathematics into a non-commutative domain:

Commutative	Noncommutative
measure space	von Neumann algebra
locally compact space	C∗- algebra
vector bundle	finite projective module
complex variable	operator on a Hilbert space
real variable	sefadjoint operator infinitesimal compact operator
range of a function	spectrum of an operator
K-theory	K-theory
vector field	derivation
integral	trace
closed de Rham current	cyclic cocycle
de Rham complex	Hochschild homology
de Rham cohomology	cyclic homology
Chern character	Chern-Connes character
Chern-Weil theory	noncommutative Chern-Weil theory
elliptic operator	K-cycle
spin Riemannian manifold	spectral triple
index theorem	local index formula
group, Lie algebra	Hopf algebra, quantum group
Symmetry	action of Hopf algebra

In the non-commutative domain (quantum mechanics) one encounters a universal definition of a time flow which has no counterpart in commutative mathematics (or classical physics). In this sense time is a mathematical necessity of quantum mechanics arising out of operator non-commutativity.

But the implication works in the other way too. Starting with the necessity of time, we can consider infinitesimal time evolution and extract the Leibniz identity. And in the categorical approach or quantum mechanics reconstruction the Leibniz identity is the main starting point.

Next time I'll expand on this and the implication for the measurement problem.

The pictorial formalism of quantum mechanics

Today I want to talk about a nice formalism of quantum mechanics developed by Samson Abramsky and Bob Coecke. The underlying mathematical formalism is that of category theory. 

The pictorial formalism describes systems and processes. The best analogy (with very good reason) is with computer science. There FORTRAN was one of the earlier languages of functional programming. In Fortran one writes functions which take an input, perform some transformation (does a computation) and generate an output. One can formally represent such a program with a box connected by two wires: the input and the output. From a high level perspective it does not matter what goes on inside the box. More important than the inner workings is that those functions can be executed one after another and can be combined like Lego to generate complexity.

Similarly in the pictorial formalism one encounters linear transformations of the wavefunction and those transformations can be combined for complexity. What makes it all interesting is that one can operate simultaneously on several inputs (like on two particles in a singlet state), or on parts of composite quantum systems.

The pictorial rules are extracted from the usual Hilbert space formulation to guarantee agreement with quantum mechanics standard computations. The Choy-Jamilkowsky isomorphism is baked in from the beginning in the approach. Here are some primitive concepts. A state is a process with no input and one output, and a test (measurement) is a process with an input and no output:

The combination of a state followed by a test gives you the probability. 

Other primitive notions are tensor product \(\otimes\) and composition \(\circ\):

\(f \otimes g = f~~while~~g\)

\( f \circ g = f~~after~~g\)

A fundamental relationship is this:

\((g_1 \otimes g_2) \circ (f_1 \otimes f_2) = (g_1 \circ f_1)\otimes (g_2 \circ f_2)\)

which can be proven pictorially by inspecting this diagram:

(g1 while g2) after (f1 while f2) = (g1 after f1) while (g2 after f2)

Quantum information is trivially represented in this approach. For example here is how teleportation protocol is drawn:

Alice shares with Bob a Bell pair (the bottom triangle which represent a state) and Alice performs a bipartite measurement on the qubit to teleport (the leftmost line) and one of the Bell particle (the upper triangle). Then she transfers a classical bit to Bob who can use it on his half Bell pair particle to recover Alice's original qbit. The information flow can be continuously traced as in the line on the right. Sometimes the information seem to flow from the future to the past, but the line can be deformed by pulling the ends to straighten it and restore the causal order. 

My poor drawings in Paint do not do justice to this very powerful method to represent the information flow. In explaining this approach I have a major weaknesses: I do not know the latest standardization of symbols. However I understand that a book on the pictorial formalism will soon become available and this will clarify the notation.

In the meantime I want to encourage the reader to look up for themselves this amazing approach. Here is an old but good reference:

http://arxiv.org/pdf/0908.1787v1.pdf

In the pictorial approach complex quantum computations becomes child's play. One can even compare this method with Feynman's diagrams in terms of simplification of computation. 

I said in prior posts that I was not aware a few years ago that my research area of reconstruction of quantum mechanics from physical principles is categorical in nature. But I can now take it a step further and create a pictorial proof of my results (it looks like a tangled mess so I won't draw it here). For this I need to introduce the concept of products and coproducts. In general any product (e.g. complex number multiplication, group operation, etc.) can be understood as a machine which takes two elements and generate another element. Flip the machine around and you get the related concept of a coproduct:

Because quantum mechanics is universal, it applies just as well to single physical systems and to composite systems which are represented using the tensor product ["while", \(\otimes\)]. It is the interplay between products, coproducts and the tensor product which completely determines the algebraic structure of quantum mechanics. All I need is a basic starting point: a product which appears naturally. And this is the Leibniz identity which comes out of the fact that the laws of nature are stable and unaffected by the passage of time. In the infinitesimal case this generates the Leibniz identity which is nothing but the good old fashion product rule of differentiation. Two mathematical representations of this product are the commutator and the Poisson bracket and they correspond to the quantum mechanics Hilbert space and phase space formulations. But I will talk more about this in future posts. 

UPDATE:

I was asked on Twitter the following excellent question:

"Has use of this pictorial/categorical formalism led to any new results? (as Feynman diagrams certainly did)"

I do not want to reveal without permission the identity of the person who asked this question, but I want to give here an extended reply which was not possible under Twitter's insane character limit.

My answer is that the pictorial formalism did not led to new results (as far as I know) as this formalism is strictly a reformulation of the Hilbert space formalism and what you can do in one you can do in the other one. The categorical approach proved its usefulness in quantum mechanics reconstruction in explaining why the results were the way they were, but this was hindsight, an "aha moment". The original motivation came from a very different direction: the attempt to find a common axiomatization for classical and quantum mechanics. 

On Feynman diagrams I am not sure what were the new results which came from it. And here is why I say this. When I was in grad school studying QFT from Mandl and Shaw, the professor did not follow the book for the first half of the semester but instead he forced us to use non-relativistic pre-Feynman diagrams, just to appreciate what Mr. Feynman actually achieved. The non-relativistic diagrams were painful to compute, and you have like 16 non-relativistic diagrams for one relativistic Feynman diagram, but you can actually do exactly what Feynman diagrams could, just with a lot more work.

Joy Christian's program of achieving quantum correlations with Clifford algebras

In the last post I explained how the algebra of the projector operators cannot always be Boolean, otherwise the Hilbert space formalism of quantum mechanics is invalid. Today I will stay in the classical-quantum divide area and I'll talk about an invalid proposal by Joy Christian which generated a lot of debate (and acrimony). When I attended the Vaxjo conference people looking up my archive record saw that I have argued against this proposal and I was asked to explain why it is invalid.

The story begins with the EPR-B experiment and the derivation of the correlation

\(-a \cdot b\)

between Alice and Bob when Alice orients her detection device on direction a and Bob orients his on direction b. So the corelation curve is minus the cosine of the detection angles (the blue line below):

In this experiment the two spin 1/2 particles are in a singlet state:

\( |\Psi\rangle = \frac{1}{\sqrt{2}}( |up \rangle_{left} |down \rangle_{right} - |down \rangle_{left} |up \rangle_{right} )\)

and because the observables are \(a \cdot \sigma\) and \(b \cdot \sigma\) the correlation is:

\( \langle \Psi | (a \cdot \sigma)\otimes(b \cdot \sigma) | \Psi \rangle\)

So how can we compute this? We use an identity:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma\)

which yields the final answer because \( \langle \Psi | \sigma| \Psi \rangle = 0\) as the mean value for both Alice and Bob are zero for any direction because we started with a total spin zero state (a singlet state).

Now Joy noticed this identity and thought that it would be nice if he could use it in a classical setting to recover the \(-a\cdot b \) correlation. There is a "little" problem: how to make the pesky \(i (a\times b) \sigma\) term disappear? 

So Joy came up with the following proposal: half the particle pairs obey:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma\)

and the other half obey this:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b -i (a\times b) \sigma\)

and so when averaged you get to the quantum correlation: \(-a \cdot b\)

But how can this be possible? It is all in the sign of \(\sqrt{-1}\) Joy claimed. When complex numbers are represented in a plane, the imaginary unit corresponds to the vertical axis. So for half of the particle pairs we draw the imaginary axis bottom up, and for the other half up bottom. But do we really get the cancellation? Nope because \(a\times b\) is a pseudo-vector which upon this reflection against the horizontal axis changes signs as well and the identity remains:

\( (a \cdot \sigma)(b \cdot \sigma) = -a\cdot b +i (a\times b) \sigma\)

and all of Joy's ill fated proposal is based on a "forgotten" -1 sign.

But if Joy would have presented his proposal like this, it would not have gotten very far. Instead Joy explained it all using the language of Clifford algebra which is not at all familiar to physicists. Also there was an associative faulty narrative about "topologically complete reasoning".

The main discovery of Joy was a no man's land at the intersection of math, physics, and philosophy: the mathematicians understanding Clifford algebra knew nothing of Bell, the physicists did not know how to counter Joy's philosophical narrative, and the philosophers had no clue of Clifford algebra. Add to this Joy's aggressive and patronizing defense of his proposal and you get a perfect storm of controversy.

The full story of debunking this nonsense would make for a nice soap opera. I was not the first who noticed the mathematical errors in Joy's proposal, I was the third out of four. Also I was not the first who wrote a paper about it, I was the second one out of three, but I was the first who uploaded it on the archive. There were other archive replies to Joy before me but nobody actually bothered to double check his math. The first reply by Marcin Pawlowski came very close to point out the problem but Joy's reply managed to discourage his critics into challenging his math:

"More specifically, the critics culminate their charge by declaring that, within my local realistic framework, it would be impossible to derive “a scalar in the RHS of the CHSH inequality. QED.” If this were true, then it would certainly be a genuine worry. With hindsight, however, it would have been perhaps better had I not left out as an exercise an explicit derivation of the CHSH inequality in Ref.[1]. Let me, therefore, try to rectify this pedagogical deficiency here."

And so people thought at that time that Joy is wrong, his physics and philosophical arguments were nonsense, but his math was correct and it was not a good idea to challenge him at that. But as it turned out all his math was only smoke and mirrors with more and more mathematical mistakes to cover up the prior ones, and I can write up an entire book about it.

All this controversy has hopefully came to an end with Joy resigning his FQXi membership, but he actually never accepted he was wrong and continues to this day to call his critics arguments: "strawmen arguments".

There was only one person who had more energy and spent more time than me debunking Joy's claims and this is Richard Gill

and the physics community owes him a debt of gratitude for putting this nonsense to rest. There were also two good things coming out of this challenge to Bell's theorem.

First, Sascha Vongehr came with what he called a Quantum Randi Challenge: show you beat Bell's theorem on a computer or shut up. With programming help from Cristi Stoica I came up with this simple Java Script program which runs in any web browser which anyone can use to try to disprove Bell's theorem until they really understand why it is an impossible task.

Second, James Owen Weatherall actually manage to fulfill Joy's hope to eliminate the extra term in a mathematically valid model which was not using Clifford algebras. But then would this count as a "disproof" of Bell's theorem? NO because the actual experimental outcomes are +1 and -1 and the correlations must be computed using them and not in a space of make-believe statistics. 

So even if Joy's math were valid, it would not represent in any way a "disproof" of Bell's theorem. 

Now what I found completely amazing was that after resigning his FQXi membership Joy received encouragements to continue the fight. It is not clear what fight. The fight to prove +1 = -1? Didn't I say soap opera?

Post Script: In case anyone has questions regarding any past or present mathematical, physical, or philosophical claims by Joy, feel free to ask here and I will answer. 
Post Post Script: in an online reply at Sci.Physics.Foundations, Joy claims his model it is not about the sign of sqrt(-1). Oh yes it is, by Hodge duality. And Joy knows that because he used to call the two Hodge duality for distinct Clifford algebras "Joy duality" which earned him high marks in his crackpot index for naming equations after yourself. The moderator of that blog, FreddyFizzx is in cahoots with Joy to promote Joy's ideas and suppress sane opposite points of view.

Boolean logic and quantum mechanics

In the last physics post I made the following remarks:

"In quantum mechanics sets and Boolean logic do not apply. When you measure something in quantum mechanics you project to a subspace of the Hilbert space and the Boolean logic changes to the logic of projections. When a system has a property like say spin this is not representable as a point in a set."

which prompted this reply from Lubos Motl:

"Even more obviously, it is complete nonsense - as you state - that quantum mechanics violates the laws of Boolean logic."

I was in the process to systematically rebut point for point all prior objections from Lubos, starting with Bertlmann's socks. However, the importance all of other points pale in comparison with this incorrect statement of Mr. Motl. It makes no sense to split hairs on finer points of disagreement when basic well known facts about quantum mechanics are misunderstood. 

To appreciate how blatantly incorrect is Lubos' objection, the implication that in quantum mechanics Boolean logic always applies amounts to denying the applicability of Hilbert spaces!!! Why? Because one can reconstruct the Hilbert space from orthomodularity, completeness, atomicity, and the covering property. But what does all this mean? Let's proceed.

The first thing we need to understand is that Boolean logic is a well defined mathematical structure with sharp axioms. In particular it needs to satisfy the so-called distributive property:

\(A \wedge (B \vee C) = (A \wedge B)\vee (A \wedge  C)\)

no ifs, ands, and buts. However this is not always true in quantum mechanics. A simple way to see this is by using the uncertainty principle. Let the three propositions A, B, and C be the following:

A = momentum of the particle is in between \(p_1\) and \(p_2\)
B = position of the particle is in between \(x_1\) and \(x_2\)

C = position of the particle is in between \(x_2\) and \(x_3\)

The left hand side reads:

"momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_1\) and \(x_3\)"

and the right hand side reads:

"momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_1\) and \(x_2\)
OR
momentum of the particle is in between \(p_1\) and \(p_2\) and the position of the particle is in between \(x_2\) and \(x_3\)"

So this seems to be identical, but if the \(x_1, x_2, x_3\) are close enough then they can be picked in such a way that left hand side obeys the uncertainty principle while the right hand side violates it. And therefore the distributive property can be violated by quantum mechanics.

No Distributivity Ghost of Classical Physics Allowed

After proving that Boolean logic is not applicable in quantum mechanics we need to figure out a replacement. After all, quantum mechanics is not in any way illogical. To do that we start with yes/no questions we can ask a physical system and attempt to organize it in a consistent way. Some questions are more general than others and when a general question is true, so are the particular ones. This means that we can define a partial order structure. Because we can ask nature no questions, or the trivial question of the existence of the physical system (which is always true) we have in fact a bounded lattice.

More can be said. Because the quantum logic area is rather dry, it is helpful to visualize the concepts using Hilbert spaces. The end goal is to reconstruct the Hilbert space from lattice properties, but we can start in reverse: from Hilbert space we will extract the properties of the lattices of propositions.

By the projection postulate, a measurement collapses the wavefunction to a 1 dimensional subspace of the Hilbert space corresponding to the eigenvector. Then one can decompose the Hilbert space into this subspace and its orthogonal complement. In general the complement is an involution and in terms of lattice of propositions we have what it is called an orthocomplement.

A lattice is called modular if it satisfies a weaker distributive law:

if \(a \leq c\)  then \(a \vee (b \wedge c) =  (a \vee b) \wedge c\) 

Also a lattice is called orthomodular if the modular condition holds only for b = orthocomplement of a. Therefore we have the following hierarchy:

distributivity => modularity => orthomodularity

[Exercise 1: come up with 3 propositions a, b, c about a quantum system for which modularity is violated due to the uncertainty principle. Hint: adapt the position and momenta example from above]
[Exercise 2: what happens if in your example for exercise 1 you replace modularity with orthomodularity? Why is the uncertainty principle not violated in this case?]

We still need more properties, that of completeness, atomicity, and the covering property to recover the Hilbert space. 

When for all collection of elements in a lattice we have a infimum and a supremum the lattice is called complete.

An element is called an atom if \(0 \leq a \leq p\) implies either \(a=0\) of \(a = p\). A lattice is called atomic when every nonzero element majorizes at least one atom.

We say that a covers b if \(a > b\) and \(a \geq c \geq b\) implies either \(c=a\) or \(c=b\). An atomic lattice has the covering property if for every a  and an atom p, such that \(a\wedge p = 0\), \(a \vee p\) covers a.

Now we have all the ingredients to reconstruct the Hilbert space for quantum mechanics. This was done by Constantin Piron in 1964 and the construction goes in three steps:

1. Embed the orthomodular, complete, atomic, with the covering property lattice in a projective space
2. Define an isomorphism from the projective space into a vector space
3. Restrict the vector space to a subspace corresponding to the elements of the lattice.

The details are too technical, but in the end one obtains quantum mechanics over reals, complex, and quaternionic numbers.

So, what does all of this mean?

There are several conclusions:

1. The questions we can ask a quantum system form an orthomodular, complete, atomic, and with the covering property lattice.
2. There is a description duality: [QM in Hilbert space] - [orthomodular lattice + additional properties] as each can be derived from the other. 
3. The logic of QM is that of projection operators (or subspaces in a Hilbert space) and not the Boolean logic of ordinary sets and Venn diagrams. In fact it is easy to visualize quantum logic statements using ordinary 3D space and picturing intersections and unions of points, lines, and planes. 
4. Quantum OR and Quantum NOT are distinct from Classical OR and Classical NOT. Quantum OR corresponds to the superposition principle.
5. Quantum AND is the same as Classical AND.
6. The logic of QM does not always satisfy the distributivity property of Boolean logic.

Let's illustrate the point of 3D visualization trick with the modularity condition. First, let's prove that distributivity implies modularity:

From distributivity we get that \(a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) = (a \vee b) \wedge c \)  when \(a \leq c\). The point of modularity is that this weaker distributivity holds only for \(a \leq c\) and not in general.

Now the ordinary 3D Euclidean space is endowed with an inner product and it is in fact a Hilbert space. Sure, it is not a complex Hilbert space which ordinary quantum mechanics demands, but Piron's result is equally valid for the usual complex number formulation as well as for the unusual quaternionic and real number formulations of quantum mechanics. This visualization trick works only for 3 dimensional Hilbert spaces but for pedagogical purposes it is enough to quickly reject invalid quantum logic identities. We only use it as a visual tool to illustrate quantum logic statements and build a visual intuition instead of being confined in the abstract and dry land of logic. 

Back to modularity, suppose a is a line and c is a plane which includes a. Hence  \(a \leq c\). If b is a point outside the plane c, \(b \wedge c\) is the null set and the left hand side evaluates to \(a \vee (b\wedge c) = a \vee null = a\). For the right hand side, \(a \vee b\) defines another plane which intersects the plane c precisely at a and so \( (a \vee b) \wedge c = d \wedge c = a\) as well. 

This is way cooler than flatlander's Venn diagram reasoning

Please note what \(a \vee b\) corresponds to: it is the geometric figure spanned by the elements a and b which in this case above is the yellow plane. Alternatively it is the smallest subspace containing both a and b.

You can also try to draw yourself the case when b is a line which intersects the plane c in a point and figure out what the modularity relation would correspond to in this case. [Exercise 3: do it. Hint: the answer is that both sides of the modularity equality are c in this case.]

[Exercise 4: why is classical OR different than quantum OR? Hint: Quantum OR for a and b is the smallest subspace containing both a and b. How is this different for classical mechanics? The fact that in the quantum case there are points in between a and b for "a OR b" corresponds to the existence of continuous transformations between pure state. No such thing exists in classical mechanics.]

[Exercise 5: why is classical AND the same as quantum AND? Hint: compare Venn diagram intersections with k-dimensional objects intersections.]

[Exercise 6: why is classical NOT different than quantum NOT (the orthocomplement)? Hint: imagine a quantum proposition a which is a line and a system whose state is a point not in a and not in NOT a. Picture it using the 3D method from above. Can this happen in Boolean logic?]

Why are quantum properties like spin measurement outcomes not always representable as points in a Venn diagram (which was the lesson from d'Espagnat's pedagogical simplification of Bell's theorem)? Because the questions we ask a quantum system are sometimes represented as lines, planes, etc. which have a geometric structure richer than that of a point. And why is that? Because of superposition which is the novel physical property which exists in the quantum world and is not present in the classical case.

To understand quantum mechanics you have to free yourself from thinking classically in terms of points, sets, Boolean logic, and Venn diagrams (equivalently thinking that physical systems have sharply defined (dispersion free was an old term for this) properties before measurement). This is all classical baggage unfit to describe nature. The logic of nature is far richer and has a clear geometric representation. Don't let the unfamiliar nature of complex Hilbert spaces stand in the way of your visual intuition.

[Hard Exercise 7: after playing with drawings of points, lines, planes to visualize modularity and orthomodularity, attempt to describe probabilities in quantum mechanics in the geometric representation of quantum logic. Hint: probabilities correspond to a simple geometric concept. What is it?]

Coming back to the original statement: "it is complete nonsense [...] that quantum mechanics violates the laws of Boolean logic" we see now that it is incorrect and moreover this is known to be incorrect for more than 80 years. The complete understanding of what is going on is known for more than 50 years. The quantum logic field is still an active research area. Yet again, category theory plays a key role because it provides semantic-free mathematical objects for logic, meaning it provides uniqueness proofs for logic concepts. 

What I presented today is well documented in the literature. For the interested reader who wants to go in depth, an excellent entry point resource is the Beltrametti and Casinelli book: The logic of quantum mechanics.

Impressions from Alaska

I just got back from a trip to Alaska and as I was preparing today to write a new physics post I got roped into a swim team B-meet timing activity which killed my free time. Since I don't want to delay my weekly post any longer and my remaining time will not do justice to the physics topic I want to write about, I will present my vacation impressions instead. I will return to physics topics at the end of this week.

Alaska is a beautiful and expensive place. The price of much anything is doubled because it has to be shipped in. The local economy is based mostly on oil and government employment. The tourist industry is big business too in the summer. The summers in Alaska are very rainy and cold. A temperature of 65 degrees Fahrenheit is considered a "heat wave" by the natives. The timing of the trip was rather poor as the annual salmon runs did not yet start. And without salmons, the wildlife was hard to be seen. In fact I see more wildlife in my backyard every day than I saw in one week in Alaska if I am not counting mosquitoes and bald eagles

which were very common.

There was one thing I did not know: upon entering freshwater to spawn, salmons undergo chemical transformations and their meat turns green inside making it unusable for human consumption. However grizzly bears do not mind that. The salmons in grocery stores are all caught in saltwater. 

Back to the trip, I sailed there on a southbound Princess cruise and I visited Anchorage, Skagway, Juneau, and Ketchikan. 

In Skagway I hiked on the Chilkoot pass and I also panned for gold in frigid water. It turned out that the gold flakes (about 1 millimeter square) were bought from New York at the price of $2 each and mixed with the river mud for gullible tourists to have an "authentic gold rush" experience at the excursion price of $125 per person :) 

The Klondike gold rush brought at the time an influx of 100,000 people out of which only about 500 became rich. The people who were able to suffer two or more consecutive winters in Alaska were called "sourdoughs" because the experience turned them bitter and angry (hey I know a sourdough blogger- wink wink Lubos).  

In Juneau I visited the Mendenhall glacier

Glaciers are rivers of of blue ice which flow at a rate of about 5 feet per day. In the process they make a lot of noise like thunder and small chunks of ice break up into the sea about every 5 minutes. It is extremely rare to have large icebergs formed this way.

The cruise boats go very close to the glaciers, but I don't have pictures to show as I lost my phone with all the good pictures in Skagway (but I'll get it back in a week). The up-close glacier view is truly majestic and awe inspiring. 

All in all the trip was outstanding and worth the money but I should have probably postponed it for about a month to experience the salmon runs. I did not mind seeing only two grizzly bears about a mile away, but besides the bald eagles, three birds and a squirrel was a bit too little in terms of wildlife for a week.