The realist epistemic view and the PBR theorem

Continuing with the realist psi-epistemic view discussion, let me first discuss if this paradigm deserves the attention that it got over the years. When one wants to establish independence of axioms/postulates, toy theories are a standard tool. However the explanations provided by realist psi-epistemic paradigm are far short of what mature quantum interpretations like Bohmian, MWI, GRW, etc do. So I am very puzzled by the traction Spekkens' toy theory got in the foundation community. This reminds me of the famous Alan Greenspan phrase of "irrational exuberance". Personally I've never fallen in love with this paradigm because I think the most effective way to spend my limited time and attention are on well defined problems. I do not know of any example from history where a philosophical paradigm was able to make genuine scientific advances. After new theories of nature are worked out and proposed, a new paradigm is formed to explain it, but not the other way around. It is more likely that a paradigm without results reinforces biases and prejudices than produces genuine advance of knowledge. 

Anyway, the realist psi-epistemic view has suffered a major blow from a recent theorem by Pusey, Barrett and Rudolf (the PBR theorem). To the people who were in love with the paradigm this was a big deal, to me however, it does not look important because I found the original paradigm uninteresting to begin with. So what does PBR shows? It all boils down to this picture:

In the psi-epistemic view, there is an overlap in the support of a hypothetical ontic states and a measurement simply reveals what it is hidden. So suppose there is this overlap between the red and green pure states that you see on the horizontal and vertical axis. By some clever quantum basis arguments, PBR shows that the black square cannot exist (see my prior post for the technical details) and so there must be no overlap to begin with, and so the realist epistemic point of view is invalid. But is realism who is at fault, or is the epistemic idea at fault? Here is where the PBR result generated a lot of controversy. The original title was poorly chosen: "The quantum state cannot be interpreted statistically" and it was later changed to "On the reality of the quantum state". 

If I were to venture a guess, it looks to me that Pusey, Barrett, and Rudolpf were originally realist psi-epistemic themselves, and they were unwilling to give up realism. With their poorly chosen title they drew the ire of Lubos Motl. That and most of other immediate reactions were basically only knee jerk reactions. PBR theorem is valid and is subject of celebration for the realist ontic camp like the Bohmian camp, but it is considered irrelevant by people who do not buy into the realist interpretations of quantum mechanics. 

And how about the realist psi-epistemic view? Unless it reinvents itself and produces a comprehensive interpretation of quantum mechanics, it is confined to the long list of failed ideas in physics like phogiston theory. The epistemic point of view lives on in new-Copenhagen interpretations and QBism, while the realist point of view lives on in Bohmian interpretation, but the combination seems to be doomed.

The realist epistemic point of view 

and 

Spekkens Toy Theory

There is about a week until I'll attend "The New Directions" conference in Washington DC, and after the conference I'll have plenty of fresh new ideas to present. In the meantime I'll devote this and next post to discuss the so-called Psi-Epistemic point of view of Quantum Mechanics.

The basic intuition originates from phase space where particles have a well define position and momenta, and a probability distribution in phase space corresponds to genuine lack of knowledge. In the realist epistemic point of view, the wavefunction corresponds to knowledge about an underlying ontic reality. This ontic reality is left unspecified: it could be classical physics with hidden variable, it could be the wavefunction itself, or it could be something completely new and undiscovered.

The key question is this: can the ontic state exist in more than one epistemic state? If yes, then a measurement in quantum mechanics simply reveals the pre-existing reality. There are a lot of roadblocks to construct such an epistemic model, but the point of view taken by Spekkens was different: let's not construct a model which recovers completely quantum mechanics predictions, but let's construct a simple epistemic toy theory and see what unintuitive quantum phenomena get a simple explanation.

The basic idea is that of simulating spin 1/2 particle measurements on 3 axis: x, y, z. Here is a picture from the excellent review paper by Matt Leifer: http://arxiv.org/pdf/1409.1570v2.pdf 

In Spekkens toy model there are 2 x states: + and - and 2 y states: + and -. The system is at any point in one of the 4 possible states: ++, +-, -+, --, but here is the catch: you cannot measure both at the same time.  Moreover, during measurement the particle makes a jump from the unmeasured state to the other. 

The spin x measurement corresponds to measuring the x coordinate, the spin y measurement correspond to measuring the y coordinate, while the measurement of spin z corresponds to measuring the "sameness of x and y" coordinates.

Repeatable measurements always yields the same outcome, just as in the quantum case, and measurement of "spin x" followed by a measurement of "spin y" perturbs the system (remember the jumping of the unmeasured coordinate during measurement) and a third measurement of "spin x" gives a random outcome. 

Now here are the successes of the toy model:

• nonorthogonal pure states cannot be perfectly distinguished by a single measurement
• no-cloning
• non-uniqueness of decomposition of mixed states

Given such impressive successes a lot of people in the quantum foundations fell in love with the realistic epistemic point of view. No full blown realistic epistemic model for quantum mechanics was ever developed, and the PBR theorem which I'll talk about next time crushed any hopes for it (or so is my opinion). 

Of course there is the other possibility of having a non-realistic epistemic interpretation, like Copenhagen and neo-Copenhagen and this possibility is alive and well.   

The number systems of quantum mechanics

Now we have reached the end of the series of the number system for quantum mechanics. Quantum mechanics can be expressed over any real Jordan algebras (including spin factors), but which one is picked by nature? The simplest case is complex quantum mechanics because you can construct the tensor product and the number system is commutative. There is a theorem by Soler which restricts the number system to real numbers, complex numbers, and quaternions but the starting assumptions are too restrictive. There is no need to force the inner product to generate only positive numbers.

When the number system is the real numbers, then this can exists only as an embedding in complex quantum mechanics so we want to build out number system from matrices of complex numbers and quaternions. The existence of the tensor product is not a requirement in general. Two fermions together do not form another fermion. But what is the meaning of a number system beside complex numbers? Basically this adds internal degrees of freedom. Do we know of additional degrees of freedom? Yes. They are the gauge symmetries.

The natural framework for discussing the number system for quantum mechanics is Connes' spectral triple. The number system is the algebra \(A\) in the spectral triple, while the unitary time evolution or the Zovko equation of continuity for quantions gives rise to the Dirac operator \(D\) in the spectral triple. The standard model arises in this formalism by a judicious pick of the algebra which gives the internal degrees of freedom. The selection of \(A\) is now done ad-hoc to simply recover the Lagrangian of the Standard Model.

One may imagine different universes where the algebra is different. Quantionic quantum mechanics does not describe our universe because we do not see a long distance Yang-Mills field with the gauge group SU(2)xU(1). Instead the electroweak field is subtler and there is a mixing of a U(1) with SU(2)xU(1) with the Weinberg angle so you may say that our universe resulted in part from a coupling between complex and quantionic quantum mechanics.

Still, regardless of the number system picked by nature for quantum mechanics, everything reduces to complex quantum mechanics when the internal degrees of freedom are ignored. This is because there are only two number systems which respect the tensor product, and in the non-relativistic limit they are identical. In complex quantum mechanics, the additional degrees of freedom form superselection domains, and C* algebras are compatible with superselection rules.

Quantionic Quantum Mechanics and Dirac's Theory of the Electron

Now we will present the relationship between quantions and spinors. They are basically two methods of taking the square root of the d'Alembertian: while spinors work in any dimensions, quantions are related to Hodge decomposition and this works only in 4 dimensions because of the interplay between 2-forms and their dual.

But let's start with the beginning. Nikola Zovko from the Ruder Boskovic institute in Croatia was following Emile Grgin's work very closely and he wanted to relate this work with known physics. To this aim he proposed an interpretation for quantionic quantum mechanics, an interpretation which generalizes Born rule: instead of probabilities the inner product will produce a 4-vector probability density current. Grgin calls this "the Zovko interpretation" and everything follows from it. In the regular complex number quantum mechanics, the wavefunction of say the electron in the hydrogen atom attaches to each point in space a complex number. Now in quantionic quantum mechanics each point in space time has attached a quantion and we know from last time that \(q^{\dagger} q\) (the "algebraic norm") is a future-oriented 4-vector. Summing over all complex number or quantion algebraic norms over the entire space yield either a positive scalar or a future oriented 4-vector and this is the Born rule. For quantions if  \(q^{\dagger} q = j\) is a 4-vector current then we must have an equation of continuity:

\({\partial}_{\mu} j^{\mu} = {\partial}_{\mu} (q^{\dagger} q)^{\mu} = 0\)

So now suppose we have a "quantionic field": \(q(x) = (q_1 (x), q_2 (x), q_3 (x), q_4 (x))\) with x the usual 4-vector in relativity. Then the continuity equation can be written as:

\({\partial}_{\mu} j^{\mu} = \frac{1}{2} [q^{\dagger} D(q) + {D(q)}^{\dagger} q ]= 0\)

where 

\( D = \left( \begin{array}{cc} \partial_0 + \partial_3 & \partial_1 + i \partial_2 \\ \partial_1 - i \partial_2 &  \partial_0 - \partial_3 \end{array}\right)\)

and so the real part of \( q^{\dagger} D q\) must vanish. If we split \(D q\) into:

\(D q = i H q + i A q\)

with H hermitian and A anti-hermitian matrices and we interpret H as outside potential, for a free particle we have: D q = -iAq and "A" can be expressed as:

\(A = m e^{i\psi} [cos \theta \gamma^1 + sin \theta cos \phi \gamma^3 + i sin \theta sin \phi \gamma^0 \gamma^5]\)

 This is more generic than the usual Dirac's equation because quantionic quantum mechanics describe a SU(2)xU(1) gauge theory. If we restrict however to the case of \(A = m \gamma^1\) we recover completely Dirac's theory. In this case there is a one-to-one correspondence between the 4 quantionic components \(q\)s and Dirac's spinors \(\Psi\)s:

\( q = \left( \begin{array}{c} q_1 \\ q_2 \\ q_3 \\ q_4  \end{array}\right) = \sqrt{2} \left( \begin{array}{c} -\Psi_2 \\ {\Psi}_3^{*} \\ \Psi_1 \\{\Psi}_4^{*} \end{array}\right)\)

\( \Psi = \left( \begin{array}{c} \Psi_1 \\ \Psi_2 \\ \Psi_3 \\ \Psi_4  \end{array}\right) = \frac{1}{\sqrt{2}} \left( \begin{array}{c} q_3 \\ -q_1 \\ q_2^* \\q_4^* \end{array}\right)\)

and the quantionic current is Dirac's current:

\(j^{\mu} = {(q^{\dagger} q)}^{\mu} = \Psi^{\dagger} \gamma^0 \gamma^{\mu} \Psi\)

But how come nobody else noticed an SU(2)xU(1) gauge theory before? Actually... this was discovered independently by David Hestenes. 

David Hestenes

He calls it: the spacetime algebra. Quantionic algebra is nothing but the spacetime algebra. Next time we'll talk about the physics of quantionic quantum mechanics and see to what degree it can represent nature.