Quantum mechanics and unitarity (part 2 of 4)

When talking about measurement, one talks about the collapse postulate. Let us take a look of what happens with the underlying Hilbert space. During collapse, the dimensionality of the Hilbert space is reduced to the dimensionality of the subspace where the wavefunction is projected to. A key point is that the dimensionality of a Hilbert space is its sole characteristic.

Measurement is initiated by first doing the tensor product of the Hilbert space of system wavefunction with the Hilbert space of the measurement apparatus. This operation increases the dimensionality of the original Hilbert space. Then the collapse decreases the dimensionality.

As an abstract operation, the tensor product respects the properties of a commutative monoid. Short of the existence of an inverse element, this is almost a mathematical group http://en.wikipedia.org/wiki/Group_(mathematics).

To model the collapse in a fully unitary way (and free of interpretations) we would like to construct the tensor product group from the tensor product commutative monoid. Is such a construction possible? Indeed it is and it is called the Grothendieck group construction http://en.wikipedia.org/wiki/Grothendieck_group Let us explain this using a simple challenge: let’s construct the group of integers Z starting from the abelian monoid of natural numbers N. We would need to introduce negative integers using only positive numbers!!! At first sight this seems impossible. How can such a thing be even possible? N and by itself is not enough, but with the addition of an equivalence relationship it can be done.

So consider a Cartesian product NxN and we would call the first element a positive number, and the second element a negative number: p = (p,0)  n = (0, n) We would like to do something like this (p,0)+(0,n) = (p, n) = p-n

Also : (0,-q) = (q, 0) All this works in general, but the definition of a Z number is no longer unique. For example: 7=(7,0) =(8,1)=(9,2)=… and -3=(0,3)=(1,4)=(2,5)=…

Therefore we need an equivalence relationship such that two pairs (a,b) and (p,q) are considered equivalent if a+q=b+p Notice that in the equivalence relationship we used only the “+” operation of the original monoid N. The formal definition of the equivalence relationship is slightly more complex due to the need to prove the transitivity property of an equivalence relationship. We call two pairs equivalent: (a,b)~(p,q) if there is a number t such that a+q+t =b+p+t

Now since Grothendieck construction is categorical (universal), it can be applied to the tensor product commutative monoid and this will explain the collapse postulate in a pure unitary way. Please stay tuned for part 3.

Quantum mechanics and unitarity (part 1 of 4)

I will start a sequence of posts showing why quantum mechanics demands only unitary time evolution despite the collapse postulate and how to solve the problem. For reference, this is based on http://arxiv.org/abs/1305.3594

The quantum mechanics reconstruction project presented in http://arxiv.org/abs/1303.3935 shows that in the algebraic approach, the Leibniz identity plays a central and early role. But what is the Leibniz identity? It is the chain derivation rule: D(fg) = D(f) g + f D(g).

All standard calculus follows from this rule. For example using recursion one proves D(X^n) = n X^(n-1) and form this and the Taylor series, the derivation rules for all usual functions follow.

In the algebraic formalism of quantum mechanics, the Leibniz identity corresponds in the state space to unitarity. Any breaking of unitarity means that Leibnitz identity is violated as well. This is the case for example in the epistemological interpretation of the wavefunction where the collapse postulate is understood as simply information update. However (and here is the big problem), breaking the Leibniz identity destroys the entire quantum mechanics formalism. In other words, any non-unitary time evolution is fatal for quantum mechanics.

So how can we understand the collapse postulate? Is quantum mechanics inconsistent? Should quantum mechanics be augmented by classical physics to describe the system and the measurement apparatus? From http://arxiv.org/abs/1303.3935 we know that there cannot be any consistent classical-quantum description of nature. Also the formalism which highlighted the problem shows the way out of the conundrum. Part 2 of the series will present preliminary mathematical structures which will be used to show how quantum mechanics can be fully unitary even during measurement.

New Directions in the Foundations of Physics Conference in Washington DC 2013 (part 5)

“Lagrangian-Only Quantum Theory” by Ken Wharton

What I liked about Ken’s talk was the novelty of his idea. Working in the quantum mechanics reconstruction area I always ask the “what if” questions: what if the universe would obey different kinds of mathematical relationships? So what if there are no dynamical equations, not even stochastic ones? This is the premise of Ken’s approach in arXiv:1301.7012.

As an inspiration one can pick the example of statistical mechanics and work out along the following lines:

1.      Consider all possible microstates

2.      Eliminate inconsistent states

3.      Assign an equal a priori probably

4.      Calculate probabilities as Bayesian updates

Now in nature there are distinguished mathematical structures, including the existence of Hamiltonian equations of motion. If we do not consider the dynamics, we can open the door for all kinds of other kinds of ontologies, like for example simulated virtual realities. By killing the dynamics Ken is basically going outside the realm of physics in hope to discover new insights. If I understood him well in a private conversation after the talk, Ken’s approach is basically opposite that of t`Hoofts: start with a chaotic system in the IR domain and arrive at quantum mechanics in the UV area. In the process Ken claims he recovers something that asymptotically becomes Born’s rule. If correct, this would represent a genuine new insight into the origin of Born’s rule besides the ones from Gleason theorem or Zurek’s program.

One may object that going outside physics is a fool’s errand, but then we are reminded by the unphysical PR-boxes which proved fruitful into better understanding quantum mechanics. Will Ken’s approach prove as fruitful? Only time will tell and I wish him luck.

One last word about this series of posts from the conference. Here is the link to the conference page http://carnap.umd.edu/philphysics/conference.html where one can find all the brief descriptions of the talks. 

New Directions in the Foundations of Physics Conference in Washington DC 2013 (part 4)

“Quantum information and quantum gravity” by Seth Lloyd

A thought provoking talk at the conference was that of Seth Lloyd. He showed how to derive Einstein’s general relativity equation from quantum limits for measuring space-time geometry and an additional black hole assumption.

One way to think of measuring the geometry of space-time is to think of a comprehensive GPS system. Measuring time amounts to measuring the number of clock ticks and this requires energy. Everybody is familiar with the position-momentum uncertainty principle, but the energy-time uncertainty principle is not so clear cut. This is because in quantum mechanics time is a parameter, and not an operator, and care has to be exercised in interpreting the energy-time uncertainty principle.

Margolus and Levitin had obtained a bound of quantum evolution time in terms of the initial mean energy of the system E: E delta t >= hbar pi/2. From this the total possible number of clock ticks in a bounded region of space time (of radius r and time span t) cannot exceed 2Et/pi hbar. In principle, quantum mechanics does not limit the accuracy for measuring time, and all you need is to do is add enough energy. But in general relativity, adding energy in a bounded region will eventually lead to the creation of a black hole. So here is a general relativity assumption: we want the radius of the bonded region to be larger than the Schwarzschild radius Rs = 2GM/c^2

From this (in terms of the Plank time Tp and Plank distance Lp) one obtains the maximum number of clock ticks achievable in a bounded region of space time before creating a black hole: r t / pi Lp Tp

Now r*t is an area and naive field theory would suggest r^3 t. Also naïve string theory would suggest at first sight r^2 t.

From those kinds of area considerations, Seth was able to deduce general relativity equations inspired in part by Ted Jacobson’s ideas (in fact Seth collaborated with Ted on this result). Now you may ask (as I certainly did) if you start with Schwarzschild’s radius and you derive Einstein’s equations, are you not vulnerable to charges of circularity? Perhaps, but the result is still interesting.

(I have one more story to tell from the conference. Please stay tuned for part 5-the last one.)

New Directions in the Foundations of Physics Conference in Washington DC 2013 (part 3)

“What is the alternative to quantum theory” by John Preskill

As promised, here is the second story from the subsequent discussions following Preskill’s talk.

After the talk I was listening in a discussion between John Preskill and Chris Fucks and at some point John asked the question: “is there anything else besides classical and quantum mechanics?” Later in the day I approached John and I told him that I know the answer to his question and I started to present my ideas captured in http://arxiv.org/abs/1303.3935 The basic idea is simple. Suppose I have on my left side a physical system A subject to the laws of nature. Also suppose I have on my right side a physical system B subject to the same laws of nature. Then if I perform the tensor product composition of system A with system B, I would get the larger system “A tensor B” subject to the same laws of physics. From this I can extract very hard constraints on the allowed form of the laws of nature. In fact it can be shown that there are only 3 such consistent solutions. One is an “elliptic” solution which corresponds to quantum mechanics, one is a “parabolic” solution which corresponds to classical mechanics, and there is a third “hyperbolic” solution which corresponds to something we do not fully understand at this time. Another way to look at the 3 solutions is by Plank’s constant: positive, zero, or imaginary.

Mathematically the whole thing can be naturally expressed in terms of category theory, and physically it corresponds to the invariance of the laws of nature under tensor composition. Now when I explain this to different people, I usually get a polite nod followed by a polite excuse to end the discussion. However, I did not get this reaction from Preskill who asked cogent clarification questions. Also he told me to look up the recent preprint of an Anton Kapustin. I did not remember the name, and the next day I asked John to type it for me on the archive search field and lo and behold I found out this preprint: http://arxiv.org/abs/1303.6917 titled: “Is there Life beyond QM?” Now the core inspiration for my result was a 70s paper by Grgin and Petersen: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103900192 and Kapustin had the same inspiration. Reading his preprint it struck me that we had independently discovered the same thing and I only managed to upload my preprint 11 days before him. Then the mystery of John’s reaction evaporated. Preskill is colleague with Kapustin at Caltech.

So now I had good news and bad news. The good news was that I am right and my credibility got a boost. The bad news is that I have competition in an area where I thought I worked alone. When I uploaded my preprint, I left out a piece of it, related to the unitary realization of the collapse postulate. So I rushed to package this result as a separate paper and I uploaded a few days after the conference: http://arxiv.org/abs/1305.3594

The problem is that any violation of unitarity is fatal to QM as shown by the QM reconstruction project. This includes the collapse during measurement even though it can be interpreted as Bayesian information update. There is an easy remedy to this though suggested by this composability/category theory formalism and it is based on the Grothendiek group construction. (I’ll explain how this works in detail in a subsequent post). As a side benefit this solves the measurement problem and eliminates the MWI interpretation as well.

The QM reconstruction projects is an area coming of age and Luca Bombelli maintains a page keeping tabs on all such projects: http://www.phy.olemiss.edu/~luca/Topics/qm/axioms.html I believe that such approaches will eventually lead to the elimination of all known QM interpretations as QM will be just as easily and naturally be derivable as special relativity. After all, how many conferences dedicated to the “correct” interpretation of special relativity and “ict-imaginary time” do you know?

New Directions in the Foundations of Physics Conference in Washington DC 2013 (part 2)

I continue my presentation of talks from the conference. Today I present:

“What is the alternative to quantum theory?” by John Preskill

Now this is a topic front and center in my area of research and I got very excited about this. Also I have two stories to tell. The first one was from the talk, and the second one from the private discussions.

I don’t think John Preskill needs an introduction, but if you are wondering who he is, it will suffice to say he won the bet against Stephen Hawking on the information loss in black holes.

His talk was around potential experimental tests of quantum mechanics in various settings. For example people play the “big cats” game, or who has the largest “Schrodinger’s cat” in interference experiments pushing the limits with bucky balls, atomic ensembles, uranium atoms, opto-mechanical experiments. Then there is the perennial question of violations of unitarity by gravity. In case you wonder, back holes do not necessarily destroy unitarity as Hawking initially thought he proved, and the best way to see that is by counterexamples from string theory. Sure, they may not be realistic, but only one counterexample is enough to prove Hawking’s initial analysis was faulty. Can unitarity be violated under any circumstances then? This is a different discussion and I’ll cover it later.

Then Preskill talked about attempts to “tweak” the Standard Model maybe by a GRW-type mechanism and he also talked about the renormalization group and the possible emergence of the Standard Model in the infrared domain from chaos in the short distance limit. All those considerations justify the need for experimental tests. We can and should test linearity, unitarity, microcausality, Poincare invariants, gauge invariance and general covariance.

From the theoretical side, questioning those aspects looks like heresy, but from the experimental side we do want to push the limit on testing their validity. For me, if there is one principle I don’t fully understand at this time, it is microcausality (there is a big difference between knowing and understanding). There are reasons to doubt it from string theory. The way to think about this is due to space-time fluctuations, maybe microcausality is not exact but asymptotically exact. Who knows, maybe it decays exponentially. What you really want to know is: “is the property fundamental and exact, or it is only a good approximation in a certain validity range?”.

The answer to those kinds of questions can also come from reconstruction projects, like the project of reconstructing (or deriving) quantum mechanics from physical principles. To this I know the answer and I had a very interesting discussion with John Preskill later in the day. I’ll tell this story in a subsequent post. Please stay tuned.

New Directions in the Foundations of Physics Conference in Washington DC 2013

This year I attended again the “New Directions” conference and yet again there were many interesting talks. I will gradually present what I learned from this year’s conference, with the caveat that I cannot be comprehensive and cover all talks.

Landauer’s Principle

One talk was by James Ladyman on his refutation of John Norton’s rejection of Landauer’s principle.

Landauer’s Principle states that erasure of information in an irreversible computation creates heat (http://en.wikipedia.org/wiki/Landauer's_principle). In prior years, I heard Norton’s talk on rejecting Landauer’s Principle based on his one molecule gas analysis. Refuting Norton’s analysis, Ladyman presented his own arguments, and since Norton was present in the audience, the talk got really interesting in the Q&A session when Norton answered back.

Norton started with a funny joke about a recent Nature paper on this: two identical pieces of paper one which had written a long series of 1s and the other which had a random series of 0 and 1s were burned and the one with the random number generated more heat.

Then he proceeded to find holes in Ladyman’s arguments and rehashed his one molecule gas analysis (which I recalled from prior years). This time there was vigorous disagreement between the audience and Norton was on what constitutes a thermal equilibrium process for a one molecule gas. Basically, Norton’s core argument was that the piston pressing down of a one molecule gas has to be very light and hence it will have violent position fluctuations. In turn this makes a slow compression of the piston impossible and this killed Ladyman's counterarguments.

However, to me the one-molecule gas arguments look flimsy because the arguments hinge on the validity of the thermodynamic limit. And this limit was not proven to my satisfaction. It is highly likely Norton’s arguments will turn to be an artifact of this limit.

Since I am no expert in Landauer’s Principle, I cannot say with certainty who is right and who is wrong. But if I were to bet I’ll put my own money on Landauer’s principle.