Sunday, August 6, 2017

The origins of gauge theory


After a bit of absence I am back resuming my usual blog activity. However I am extremely busy and I will create new posts every two weeks from now on. I am starting now a series explaining gauge theory and today I will start at the beginning with Hermann Weyl's proposal.


In 1918 Hermann Weyl attempted to unify gravity with electromagnetism (the only two forces known at the time) and in the process he introduce the idea of gauge theory. He espouse his ideas in his book "Space Time Matter" and this is a book which I personally find hard to read. Usually the leading physics people have crystal clear original papers: von Neumann, Born, Schrodinger, but Weyl's book combines mathematical musings with metaphysical ideas in an unclear direction. The impression I got was of a mathematical, physical and philosophical random walk testing in all possible ways and directions and see where he could make progress. He got lucky and his lack of cohesion saved the day because he could not spot simple counter arguments against his proposal which could have stopped him cold in his tracks. But what was his motivation and what was his approach?

Weyl like the local character of general relativity and proposed (from pure philosophical reasons) the idea that all physical measurements are relative. I particular, the norm of a vector should not be thought as an absolute value, but as a value that can change at various point of spacetime. To compare at different points, you need a "gauge", like a device used in train tracks to make sure the train tracks remained at a fixed distance from each other. Another word he used was "calibration", but the name "gauge" stuck.

So now suppose we have a norm \(N(x)\) of a vector and we do a shift to \(x + dx\). Then:

\(N(x+dx) = N(x) + \partial_{\mu}N dx^{\mu}\)

Also suppose that there is a scaling factor \(S(x)\):

\(S(x+dx) = S(x) + \partial_{\mu}S dx^{\mu}\)

and so to first order we get that N changes by:

\(( \partial_{\mu} + \partial_{\mu} S) N dx^{\mu} \)
Since for a second gauge \(\Lambda\), \(S\) transforms like:

\(\partial_{\mu} S \rightarrow \partial_{\mu} S  +\partial_{\mu} \Lambda \)

and since in electromagnetism the potential changes like:

\(A_{\mu}  \rightarrow A_{\mu} S  +\partial_{\mu} \Lambda \)

Weyl conjectured that \(\partial_{\mu} S = A_{\mu}\).

However this is disastrous because (as pointed by Einstein to Weyl on a postcard) it implies that the clocks would change their frequencies based on the paths they travel (and since you can make atomic clocks it implies that the atomic spectra is not stable).

Later on with the advent of quantum mechanics Weyl changed his idea of scale change into that of a phase change for the wavefunction and the original objections became mute. Still more needed to be done for gauge theory to become useful.

Next time I will talk about Bohm-Aharonov and the importance of potentials in physics as a segway into the proper math for gauge theory. 

Please stay tuned.

Monday, July 10, 2017

The main problem of MWI is the concept of probability


Now it is my turn to present the counter arguments against many worlds. All known derivations of Born rule in MWI have (documented) issues of circularity: in the derivation the Born rule is injected in some form or another. However the problem is deeper: there is no good way to define probability in MWI.

Probability can be defined either in the frequentist approach as limit of frequency for large trial numbers, or subjectively as information update in the Bayesian approach. Both those approaches are making the same predictions. 

It is generally assumed by all MWI supporters that branch counting leads to incorrect predictions and because of this the focused is changed on subjective probabilities and the "apparent emergence" of Born rule. However this implicitly breaks the frequentist-subjective probability relationship. The only way one can use the frequentist approach is by using branch counting. Let's have a simple example.

Suppose you work at a factory which makes fair (quantum) coins which land 50% up and 50% down. Your job is quality assurance and you are tasked with finding the defective coins. Can you do your job in a MWI quantum universe? The only thing you can do is to flip the coin many times and see if it lands about 50% up and 50% down. For a fair coin there is no issue. However for a biased coin (say 80%-20%) you get the very same outcomes as in the case of the fair coins and you cannot do your job.



There is only one way to fix the problem: consider that the world does not split in 2 up and down branches, but say in 1 million up and 1 million down branches. In this case you can think that in the unfair case the world splits in 1.6 million up worlds, and 400 thousand down worlds. This would fix the concept of probability in MWI restoring the link between frequentist and subjective probabilities, but this is not what MWI supporters claim. Plus, this has problems of its own with irrational numbers and the solution is only approximate to some limit of precision which can be refuted by any experiment run long enough.

So to boil the problem down, in MWI there is no outcome difference in case of a fair coin versus an unfair coin toss: in both cases you get an "up world" and a "down world". Repeating the coin toss any number of times does not change the nature of the problem in any way. Physics is an experimental science and we test the validity of the theories against experiments. Discarding branch counting in MWI is simply unscientific

Now in the last post Per argued for MWI. I asked him to show what would happen if we flip a fair and an unfair coin three times to simply run through his argument on an elementary example and not hid behind general equations. After some back and forth, Per computed the distribution \(\rho\) in the fair and unfair case (to match quantum mechanics predictions) but the point is that \(\rho\) must arise out of the relative frequencies and not be computed by hand. Because the relative frequencies are identical in the two cases \(\rho\) must be injected by a different mechanism. His computation of \(\rho\) is the point where circularity is introduced in the explanation. If you look back in his post, this comes from his equation 5 which is derived from equation 3. Equation 3 assumes Born rule and is the root cause of circularity in his argument. Per's equation 7 recovers the Born rule in the limit case after assuming Born rule in equation 3 - q.e.d.

Sunday, June 25, 2017

Guest Post defending MWI


As promised, here is a guest post from Per Arve. I am not interjecting my opinion in the main text but I will ask questions in the comments section.

Due to the popularity of this post I am delaying the next post for a week.

The reason to abandon the orthodox interpretation of quantum mechanics is its incompleteness. Bohr and Heisenberg refused the possibility to describe the measurement process as a physical process. This is encoded in Bohr's claim that the quantum world cannot be understood. Such an attitude served to avoid endless discussions about the weirdness of quantum mechanics and divert attention to the description of microscopic physics with quantum mechanics. Well done! A limited theory is better than no theory.

But, we should always try to find theories that in a unified way describes the larger set of processes. The work by Everett and the later development of decoherence theory by Zeh, Zurek and others have given us elements to describe also the measurement process as a quantum mechanical process. Their analysis of the measurement process implies that the unitary quantum evolution leads to the emergence of separate new "worlds". The appearance of separate "worlds" can only be avoided if there is some mechanism that breaks unitarity.

The most well-known problem of Everett's interpretation is that of the derivation of the Born rule. I describe the solution of that problem here. (You can also check my article on the arxiv [1603.01625] Postulates for and measurements in Everett's quantum mechanics)

The main point is to prove that physicists experience the Born rule. That is by taking an outside view of the parallel worlds created in a measurement situation. The question, what probability is from the perspective of an observer inside a particular branch, is more a matter of philosophy than of science.

The natural way to find out where something is located is to test with some force and find out where we find resistance. The force should not be so strong that it modifies the system we want to probe. This corresponds to the first order perturbation of the energy due to the external potential U(x),

\(\Delta E =\int d^3 x {|\psi (x)|}^2 U(x)\)  (1)

This shows that \({|\psi(x)|}^2\) gives where the system is located. (Here, spin and similar indexes are omitted.)

The argumentation for the Born rule relies on that one may ignore the presence of the system in regions, where integrated value of the wave function absolute square is very small.

In order to have a well defined starting point I have formulated two postulates for Everett's quantum mechanics.

EQM1 The state is a complex function of positions and a discrete index j for spin etc,

\(\Psi = \psi_j (t, x_1, x_2, ...) \)  (2)

Its basic interpretation is given by that the density 

\(\rho_j (t, x_1, x_2,...) = {|\psi_j (t, x_1, x_2, ...)|}^2 \)  (3)

answers where the system is in position, spin, etc.

It is absolute square integrable normalized to one 

\( \int \int···dx_1dx_2 ··· \sum_j {|\psi_j (t, x_1, x_2, ...)|}^2 = 1\)  (4)

This requirement signifies that the system has to be somewhere, not everywhere. If the value of the integral is zero, the system doesn’t exist anywhere.

EQM2 There is a unitary time development of the state, e.g.,

\(i \partial_t \Psi = H\Psi \),

where H is the hermitian Hamiltonian. The term unitary signifies that the value of the left hand side in (4) is constant for any state (2).

Consider the typical measurement where something happens in a reaction and what comes out is collected in an array of detectors, for instance the Stern-Gerlach experiment. Each detector will catch particles that have a certain value of the quantity B we want measure.

Write the state that enter the array of detectors as sum of components that enter the individual detectors, \(|\psi \rangle = \sum c_b |b\rangle\), where b is one of the possible values of B. When that state has entered the detectors we can ask, where is it? The answer is that it is distributed over the individual detectors. The distribution is 

\(\rho_b = {|c_b|}^2 \)  (5)

This derived by integrate the density (3) over the detector using that the states \(|b\rangle\) have support only inside its own detector. 

The interaction between \(|\psi \rangle\) and the detector array will cause decoherence. The total system of detector array and \(|\psi \rangle\) splits into separate "worlds" such that the different values b of the quantity B will belong to separate "worlds".

After repeating the measurement N times, the distribution that answer how many times have the value \(b=u\) been measured is

\(\rho(m:N | u)= b(N,m) {(\rho_u)}^m{(\rho_{¬u})}^{N−m} \)  (6)

where \(b(N,m)\) is the binomial coefficient \(N\) over \(m\) and \(\rho_{¬u}\) is the sum over all \(ρ_b\) except \(b=u\).

The relative frequency \(z=m/N\) is then given by

\(\rho(z|u) \approx \sqrt{(N/(2\pi \rho_u \rho_{¬u}))} exp( −N{(z−\rho_u)}^2/(2\rho_u \rho_{¬u}) ) \)  (7)

This approaches a Dirac delta \(\delta(z − \rho_u)\). If the tails of (7) with low integrated value are ignored, we are left with a distribution with \(z \approx u\). This shows that the observer experiences a relative frequency close to the Born value. Reasonably, the observer will therefore believe in the Born rule.

The palpability of the densities (6) and (7) may be seen by replacing the detectors by a mechanism that captures and holds the system at the different locations. Then, we can measure to what extent the system is at the different locations (4) using an external perturbation (1). In principle, also the distribution from N measurements is directly measurable if we consider N parallel experiments. The relative frequency distribution (7) is then also in principle a directly measurable quantity.

A physicist that believes in the Born rule will use that for statistical inference in quantum experiments. According to the analysis above, it will work just as well as we expect it to do using the Born rule in a single world theory.

A physicist who believes in a single world will view the Born rule as a law about probabilities. A many-worlder may view it as a rule that can be used for inference about quantum states as if the Born rule is about probabilities.

With my postulates, Everett's quantum mechanics describe the world as we see it. That is what should be discussed. Not whether it pleases anybody or not.

If the reader is interested what to do in a quantum russian roulette situation, I have not much to offer. How to decide your future seems to be a philosophical and psychological question. As a physicist, I don't feel obliged to help you with that.

Per Arve, Stockholm June 24, 2017

Sunday, June 18, 2017

Impressions from FQMT 2017


Poster, FQMT

I just came back from Vaxjo where I had a marvelous time. It does sounds cliche, but this year was the best conference organized by Professor Khrennikov and I got many pleasant and unexpected surprises.

The conference did had one drawback: everyday after the official talks we continue the discussions about quantum mechanics well past midnight at "The Bishops Arms" where we drank too many beers causing me to gained a few pounds :)

At the conference I had a chance to meet and talk with Phillipe Grangier (he worked with Aspect on the famous Bell experiment) and I witness him giving the best cogent comments on all talks: from experimental to theoretical. He even surprised me when he asked at the end of my presentation why I am using time to derive Leibniz identity, where any other symmetry will do? Indeed this is true, but the drawback is that any other symmetry lacks generality later on during composition arguments. Suppose we compose two physical systems: one with with a continuous symmetry and another without, then the composed system will lack that symmetry. The advantage of using time is that it works for all cases where energy is conserved.

Grangier presented his approach on quantum mechanics reconstruction using contextuality and continuity (like in Hardy's 5 reasonable axioms paper). The problem of continuity is that it lacks physical intuition/motivation. Why not impose right away the C* condition: \(||a^* a|| = {||a||}^2\) and recover everything from it?

Bob Coecke and Aleks Kissinger book on the pictorial formalism: "Picturing Quantum Processes" was finally ready and was advertised at the conference. If you go to www.cambridge.org/pqp you can get with a 20% discount when you enter the code COECKE2017 at the checkout.

Coecke 's talk was about causal theories and his main idea was: "time reversal of any causal theory = eternal noise". This looks deep, but it is really a trivial observation: you can't get anything meaningful and you can't control signals which have an information starting point because the starting point corresponds to the notion of false and anything is derivable from false.

Robert Raussendorf from University of Vancouver had a nice talk about measurement based quantum computations where measurements are used to control the computation and he identified a cohomological framework.

One surprise talk for me was the one given by Marcus Appleby from University of Sydney who presented a framework of equivalence for Quantum Mechanics between finite and infinite dimensional cases. This is of particular importance to me as I recovered quantum mechanics in the finite dimensional case only and I am searching for an approach to handle the infinite dimensional case.

I made new friends there and I got very good advice and ideas - a big thank you. I also got to give many in person presentations of my quantum reconstruction program.

There was one person claiming he solved the puzzles of the many worlds interpretation. I sat next to him at the conference dinner and I invited him to have a guest post at this blog to present his solution. As a disclaimer, I think MWI lacks the proper notion of probability and I am yet to see a solution but I am open to listen to new arguments. What I would like to see is an explanation of how to reconcile the world split of 50-50% when the quantum probabilities are 80-20%? I did not see this explained in his presentation to my satisfaction, but maybe I was not understating the argument properly.

Saturday, June 10, 2017

Jordan-Banach, Jordan-Lie-Banach, C* algebras, and quantum mechanics reconstruction


This a short post written as waiting for my flight at Dulles Airport on my way to Vaxjo Sweden for a physics conference. 

First some definitions. a Jordan-Banach algebra is a Jordan algebra with the usual norm properties of a Banach algebra. A Jordan-Lie-Banach algebra is a Jordan-Banach algebra which is a Lie algebra at the same time. A Jordan-Lie algebra is the composability two-product algebra which we obtained using category theory arguments.

Last time I hinted about this week's topic which is the final step in reconstructing quantum using category theory arguments. What we obtain from category theory is a Jordan-Lie algebra which in the finite dimensional case has the spectral properties for free because the spectrum in uniquely defined in an algebraic fashion (things gets very tricky in the infinite dimensional case). So in the finite dimensional case JL=JLB.

But how can we go from Jordan-Banach algebra to C*? In general it cannot be done. C* algebras correspond to quantum mechanics and on the Jordan side we have the octonionic algebra which is exceptional. Thus cannot be related to quantum mechanics because octonions are not associative. However we can define state spaces for both Jordan-Banach and C* algebras and we can investigate their geometry. The geometry is definable in terms if projector elements which obey: \(a*a = a\). In turn this defines the pure states as the boundary of the state spaces. If the two geometries are identical, we are in luck. 

Now the key question is: under what circumstances can we complexify a Jordan-Banach algebra to get a C* algebra?

In nature, observables play a dual role as both observables and generators. In literature this is called dynamic correspondence. Dynamic correspondence is the essential ingredient: any JB algebra with dynamic correspondence is the self-adjoint part of a C* algebra. This result holds in general and can be established by comparing the geometry of the state spaces for JB and C* algebras.

Now for the punch line: a JL algebra comes with dynamic correspondence and I showed that in prior posts. The conclusion is therefore:

in the finite dimensional case: JL is a JLB algebra which gives rise to a C* algebra by complexification and by GNS construction we obtain the standard formulation of quantum mechanics. 

Quantum mechanics is fully reconstructed in the finite dimensional case from physical principles using category theory arguments! 

By the way this is what I'll present at the conference (the entire series on QM reconstruction).

Sunday, June 4, 2017

From composability two-product algebra to quantum mechanics

Last time we introduced the composability two-product algebra consisting of the Lie algebra \(\alpha\) and the Jordan algebra \(\sigma\) along with their compatibility relationship. This structure was obtained by categorical arguments using two natural principles of nature:

- laws of nature are invariant under time evolution
- laws of nature are invariant under system composition

What we did not obtain were spectral properties. However, in the finite dimensional case, we do not need spectral properties and we can fully recover quantum mechanics in this particular case. The trick is to classify all possible two-product algebras because there are only a handful of them. This is achieved with the help of the Artin-Weddenburn theorem

First some preliminary. We need to introduce a Lie-Jordan-Banach (JLB) algebra by augmenting the composability two-product algebra with spectral properties:
-a JLB-algebra is a composability two-product algebra with the following two additional properties:
  • \(||x\sigma x|| = {||x||}^{2}\)
  • \(||x\sigma x||\leq ||x\sigma x + y\sigma y||\)
Then we can define a C* algebra by compexification of a JLB algebra where the C* norm is:

\(||a+ib|| = \sqrt{{||a||}^{2}+{||b||}^{2}}\)

Conversely from a C* algebra we define a JLB algebra as the self-adjoint part and where the Jordan part is:

\(a\sigma b = \frac{1}{2}(ab+ba)\)

and the Lie part is:

\(a\alpha b = \frac{i}{\hbar}(ab-ba)\)

From C* algebra we recover the usual quantum mechanics formulation by GNS construction which gets for us:

- a Hilbert space H
- a distinguished vector \(\Omega\) on H arising out of the identity of the C* algebra
- a representation \(\pi\) of the algebra as linear operators on H
- a state \(\omega\) on C* represented as \(\omega (A) = \langle \Omega, \pi (A)\Omega\rangle_{H}\)

Conversely, from quantum mechanics a C* algebra arises as bounded operators on the Hilbert space.

The infinite dimensional case is a much harder open problem. Jumping from the Jordan-Banach operator algebra side to the C* and von Neuman algebras is very tricky and this involves characterizing the state spaces of operator algebras. Fortunately all this is already settled by the works of Alfsen, Shultz, Stormer, Topping, Hanche-Olsen, Kadison, Connes. 

Sunday, May 21, 2017

The algebraic structure of quantum and classical mechanics


Let's recap on what we derived so far. We started by considering time as a continous functor and we derived Leibniz identity from it. Then for a particular kind of time evolution which allows a representation as a product we were able to derive two products \(\alpha\) and \(\sigma\) for which we derived the fundamental bipartite relations.

Repeated applications of Leibniz identity resulted in proving \(\alpha\) as a Lie algebra, and \(\sigma\) as a Jordan algebra and an associator identity between them:

\([A,B,C]_{\sigma} + \frac{J^2 \hbar^2}{4}[A,B,C]_{\alpha} = 0\)

where \(J\) is a map between generators and observables encoding Noether's theorem.

Now we can combine the Jordan and Lie algebra as:

\(\star = \sigma\pm \frac{J \hbar}{2}\alpha\)

and it is not hard to show that this product is associative (pick \(\hbar = 2\) for convenience):

\([f,g,h]_{\star} = (f\sigma g \pm J f\alpha g)\star h - f\star(g\sigma h \pm J g\alpha h)=\)
\((f\sigma g)\sigma h \pm J(f\sigma g)\alpha h \pm J(f\alpha g)\sigma h + J^2 (f\alpha g)\alpha h \)
\(−f\sigma (g\sigma h) \mp J f\sigma (g\alpha h) \mp J f\alpha (g\sigma h) − J^2 f\alpha (g\alpha h) =\)
\([f, g, h]_{\sigma} + J^2 [f, g, h]_{\alpha} ±J\{(f\sigma g)\alpha h + (f\alpha g)\sigma h − f\sigma (g\alpha h) − f\alpha (g\sigma h)\} = 0\)

because the first part is zero by associator identity and the second part is zero by applying Leibniz identity. In Hilbert space representation the star product is nothing but the complex number multiplication in ordinary quantum mechanics

Now we can introduce the algebraic structure of quantum (and classical) mechanics:

A composability two-product algebra is a real vector space equipped with two bilinear maps \(\sigma \) and \(\alpha \) such that the following conditions apply:

- \(\alpha \) is a Lie algebra,
- \(\sigma\) is a Jordan algebra,
- \(\alpha\) is a derivation for \(\sigma\) and \(\alpha\),
- \([A, B, C]_{\sigma} + \frac{J^2 \hbar^2}{4} [A, B, C]_{\alpha} = 0\),
where \(J \rightarrow (−J)\) is an involution mapping generators and observables, \(1\alpha A = A\alpha 1 = 0\), \(1\sigma A = A\sigma 1 = A\)

For quantum mechanics \(J^2 = -1\). In the finite dimensional case the composability two-product algebra is enough to fully recover the full formalism of quantum mechanics by using the Artin-Wedderburn theorem.

The same structure applies to classical mechanics with only one change: \(J^2 = 0\).

In classical mechanics case, in phase space, the usual Poisson bracket representation for product \(\alpha\) can be constructively derived from above:
\(f\alpha g = \{f,g\} = f \overset{\leftrightarrow}{\nabla} g = \sum_{i=1}^{n} \frac{\partial f}{\partial q^i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q^i}\)

and the product \(\sigma\) is then the regular function multiplication.

In quantum mechanics case in the Hilbert space representation we have the commutator and the Jordan product:

\(A\alpha B = \frac{i}{\hbar}  (AB − BA)\)
\(A\sigma B = \frac{1}{2} (AB + BA)\)

or in the phase space representation the Moyal and cosine brackets:

\(\alpha = \frac{2}{\hbar}\sin (\frac{\hbar}{2} \overset{\leftrightarrow}{\nabla})\)
\(\sigma = \cos (\frac{\hbar}{2} \overset{\leftrightarrow}{\nabla})\)

where the associative product is the star product.

Update: Memorial Day holiday interfered with this week's post. I was hoping to make it back home on time to write it today, but I got stuck on horrible traffic for many hours. I'll postpone the next post for a week.