## The Quantum-Classical Debate:

Today I will reply to the arguments Andrei made last week in his guest post. But let me first start by giving out the answer to the questions from two posts ago. First, on why there is no uncertainty relationship for spin, from any two non-commuting observables A and B one obtains:

$$\sigma_A \sigma_B \geq \frac{1}{2} |\langle [A,B] \rangle|$$

The key point of Heisenberg uncertainty relationship for position and momenta is to be pedantic and observe that the commutator is proportional with the identity operator $$I$$:

$$[x,p] = i \hbar I$$

which is not true in the case of spin. Because of this, in the spin case there exists states for which the right hand side value $$|\langle [A,B] \rangle|$$ is zero. The position-momenta uncertainty remains bounded from below for any state.

For the other question on the apparent violation of the uncertainty principle. here is what Heisenberg stated:

"If the velocity of the electron is at first known, and the position then exactly measured, the position of the electron for times previous to the position measurement may be calculated. For these past times, δpδq is smaller than the usual bound." and "the uncertainty relation does not hold for the past." I think this is not a well known or appreciated fact by the majority of physics community.

Then Heisenberg pointed out that these values can never be used as initial conditions in a prediction about the future behavior of the electron.

Now back to answering Andrei's challenge to quantum mechanics, Andrei discussed 3 points:

Objection 1: Classical, local theories have been ruled out by Bell’s theorem.
Objection 2: Classical theories cannot explain single-particle interference (double slit experiment), quantum tunneling, the stability of atoms or energy quantification in atoms or molecules.
Objection 3: Even if one could elude the previous points, there is no reason to pursue classical theories because quantum mechanics perfectly predicts all observed phenomena.

Let's analyze them in turn.

On Objection 1, I agree that classical, local theories have been ruled out by Bell’s theorem with only one loophole left: super-deterministic theories pursued by 't Hooft. Any statistical theory obeying Kolmogorov's axioms respects Bell's inequalities. It is interesting to see how different realistic quantum mechanics interpretations escape Kolmogorov's axioms: Bohmian interpretation is contextual while quantum mechanics in phase space uses negative probabilities.

On superdeterminism, one needs to deny free will and this is a very tall order. While I (and anyone else) cannot give a rigorous definition of free will, I know that I have it. Andrei contends that classical theory is deterministic. While true, this is both an insufficient and an irrelevant argument. Superdeterminism is only a pre-requisite step: you need to obtain from it quantum correlations, and so far I am not aware of any successful model. Second, determinism does not imply superdeterminism because the existence of chaotic evolution equations. Predicting weather is a classical example. I do not think superdeterminism has any chance of success.

On Objection 2, I again agree with its statement. Quantum mechanics arose out of the inability of classical mechanics to explain atomic phenomena. But instead of expanding on this let's reply to the concrete arguments Andrei raised. Let's start with:

" This is all nice, but classical physics is not the same thing as Newtonian physics of the rigid body. Let’s consider a better classical approximation of the electron, a charged bullet. The slits are made of some material that will necessarily contain a large number of charged “bullets”. As the test bullet travels, its trajectory will be determined by the field generated by the slitted barrier. The field will be a function of position/momenta of the “bullets” in the barrier. But the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit, so the effect with both holes open is NOT the sum of the effects with each hole open alone."

This argument is wrong on two counts. First, one can make an interference experiment with neutrons where the neutrons not passing through the slits will be simply absorbed. Using electrically neutral particles renders irrelevant Andrei's objection. Second, "the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit" is incorrect as shown by a simple order of magnitude analysis. The electric fields near the slit are relevant on an atomic distance scale, while the distance between slits is macroscopic. You are looking at about seven order of magnitude difference in the ratio of relevant distances which translates in terms of force into a ten to minus fourteen order of magnitude effect. But the interference pattern is macroscopic and the difference between two Gaussian distributions vs. interference pattern cannot be explained away by a force fourteen of orders of magnitude smaller than what it is needed.

The next objection is appealing to Yves Couder's results. Those are interesting experiments, but they are not a confirmation of quantum mechanics emergence from classical physics and I know no one who claims it so. As such the argument is irrelevant to the current debate.

On the free fall atomic model, I did not read those papers so do not know if the claims are correct or not. The author may simply have proposed a model and analyzed the consequences and never claimed that his model describes nature.  Based on the general information available it is immediately clear that that model has nothing to do with reality. Also peer review is no magic bullet for avoiding publishing incorrect results. In my area of expertise in the last 12 months I read 2 published papers which were pure unadulterated garbage: the errors were not subtle, but blatant and packaged in a dishonest way to bamboozle the reader. Moreover I have concrete proof that the authors were aware of their mistakes when they published it. Physics is not immune to charlatans, crooks, and incompetent reviewers.

On the ionic crystals argument, without quantum mechanics the collection of electrons and nuclei will behave like a plasma and not like a crystal. This is moderately easy to test: create a computer simulation of say 1000 atoms and use Maxwell's and Newtonian equations of motion only to model the interaction. Then try to find an initial configuration which will be stable. I think there is none. Prove me wrong with such a model and I'll concede this point.

On quantum tunneling, the argument is pure handwaiving. Let me make an analogy. I know how my microwave works. But an alternative explanation might be that little Oompa-Loompas inside it are heating the food and I cannot see them because they move really fast.  The point is that the argument needs to have more predictive power than a fuzzy non-committal: "A new theory could predict a much stronger force." Show me the money. Propose such a theory and then we can discuss its merits. I am not asking something impossible. Regarding tunneling, quantum mechanics provides testable predictions which were confirmed experimentally. I only hold any alternative theory to the same level of experimental confirmation.

On Objection 3, I somewhat disagree with it. It is worthwhile to pursue non-quantum toy theories to better understand quantum mechanics, but not to search for an alternative to quantum mechanics.

Let me answer the four sub-points raised by Andrei:

a.       If nature is not probabilistic after all, there is much to be discovered. Detailed mechanism behind quantum phenomena should be revealed, bringing out a deeper understanding of our universe, and maybe new physical effects.

There is no "sub-quantum" or "hidden variable" explanation to quantum effects. Quantum mechanics is at the core of Nature, and my work is about proving this rigorously and not as a result of personal beliefs.

b.      Quantum theories are not well equipped to describe the universe as a whole. There is no observer outside the universe, no measurement can be performed on it, not even in principle.

This is a sterile objection to quantum mechanics and this cannot be used to justified a realistic alternative theory. First, even in quantum mechanics there are realistic interpretations. Second, epistemic interpretation like Qbism avoids this because they only talk about Bayesian probabilities. This objection only applies to naively using quantum mechanics in cosmology. Loop quantum gravity is unaffected by this objection.

c.       Due to its inability to provide an objective description of reality, quantum mechanics may not be able to solve the cosmological constant problem. A theory that states clearly “what’s there” could provide a much better estimate of the vacuum energy. After all we are not interested in what energy someone could find by performing a measurement on the vacuum, but what the vacuum consists of, when no one is there to pump energy into it.

The statement underscores a deep misunderstanding of the vacuum. Vacuum is actually a very violent place filled with virtual particles due to interplay of relativity and quantum mechanics. See this poor quality but brilliant video of a Sidney Coleman lecture to understand why merging them is not a trivial thing as one may naively expect that adding symmetries to quantum mechanics always results in simplifications. See also the QCD "Lava Lamp" which was shown at the 2004 Nobel Physics lecture. The cosmological constant problem is not a problem of quantum mechanics. I am not an expert in string theory but I know it has at least a solution to this problem (I don't know if it got rid of Susskind's "Rube Goldberg" label).

d.      Quantum mechanics requires an infinitely large instrument to measure a variable with infinite precision. When gravity is taken into account, it follows that local, perfectly defined properties cannot exist, because, beyond a certain mass, the instrument would collapse into a black hole.

The same argument can be used in the case of classical deterministic physics.

As you can see I disagree with the points Andrei was making, but nevertheless I want to thank him for participating in this debate and I look forward to discuss his replies in the commenting section of this post. I think such debates are useful, and I feel that the professional community of physicists is not doing a good job in engaging the public or explaining what it is doing. Physicists are very busy people trying to get ahead in a very competitive field. However the outside world usually experiences an arrogant wall of silence.

## Guest post: "Are classical, deterministic, field theories compatible with the predictions of quantum mechanics?"

Elliptic Composability is a blog dedicated to quantum mechanics and a recent post attracted a lot of attention and comments. In particular one of the readers, Andrei, has argued for classical physics and we have engaged in a lot of back and forth discussions about it. Although I disagree with Andrei, I have found his arguments well constructed and I thought my debate with him would be of interest to a much larger audience. So I have invited Andrei to present his best arguments in a guest post to which I will present my counter-arguments in the next post.

Without further ado, here is Andrei's guest post:

Thank you, Florin, for inviting me to present on your blog my arguments that classical physics can still play a role in understanding nature.

The following objections are usually raised against the search for classical explanations for quantum phenomena:

Objection 1: Classical, local theories have been ruled out by Bell’s theorem.
Objection 2: Classical theories cannot explain single-particle interference (double slit experiment), quantum tunneling, the stability of atoms or energy quantification in atoms or molecules.
Objection 3: Even if one could elude the previous points, there is no reason to pursue classical theories because quantum mechanics perfectly predicts all observed phenomena.

Objection 1
It is a widespread belief that Bell’s theorem rules out local realism. As classical field theories are both local and realistic they couldn’t possible reproduce the predictions of quantum mechanics. However, a more careful reading of the theorem would be:

Assuming the experimenters are free to choose what measurements to perform, the logical conclusion is that no local and realistic theory can reproduce the predictions of quantum mechanics.

One can easily notice that, if the motion of quantum particles (including the particles the experimenter himself is made of) is described by some classical, deterministic, field theory, this freedom of choice does not make any sense.  Classical determinism implies that certain past configuration uniquely determines the future. Anything else would require a violation of the physical law.

One can still object that, even if the experimenters (say Bob and Alice) follow some deterministic process there is still no reason to assume that their behavior would be correlated. Alice could use the decay of some radioactive material to decide the measurement she performs, while Bob could let his measurement be decided by very complex random number generation software. Any assumed correlation between such unrelated processes would amount to a conspiracy no one should seriously consider.

Let’s analyze however, this experiment from the point of view of a classical field theory. One can make the following two observations:

a.       the experiment reduces, at microscopic scale, to three groups of particles: A (Alice, her radioactive material and the detector she controls), B (Bob, his computer and the detector he controls), S (the source of the entangled particles).
b.      The trajectory of a particle in A would be a function of position/momenta of particles in A,B and C; the trajectory of a particle in B would be a function of position/momenta of particles in A,B and C; The trajectory of a particle in C would be a function of position/momenta of particles in A,B and C

From (a) and (b) it follows that Alice and Bob (and the source of the entangled particles), cannot be independent of one another. The so-called conspiracy is a direct consequence of the mathematical structure of a classical field theory.

In conclusion I have proven that Bell’s theorem cannot rule out classical deterministic field theories.

Objection 2
I will show here how some “mysterious” quantum effects are not in contradiction with what one would expect if some classical field theory describes the motion of particles at the fundamental level. I will start with the iconic double-slit experiment as it is presented by Feynman here:

Feynman says:
“In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works.”

I invite everyone to read the whole description of the experiment there. Feynman chooses to use indestructible bullets as the classic analogs of electrons. The experiment is performed with the first slit open, then with the second slit open, and, in the end, with both slits open. The expected result is:

“The probabilities just add together. The effect with both holes open is the sum of the effects with each hole open alone. We shall call this result an observation of “no interference,” for a reason that you will see later. So much for bullets. They come in lumps, and their probability of arrival shows no interference.”

This is all nice, but classical physics is not the same thing as Newtonian physics of the rigid body. Let’s consider a better classical approximation of the electron, a charged bullet. The slits are made of some material that will necessarily contain a large number of charged “bullets”. As the test bullet travels, its trajectory will be determined by the field generated by the slitted barrier. The field will be a function of position/momenta of the “bullets” in the barrier. But the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit, so the effect with both holes open is NOT the sum of the effects with each hole open alone.

In this paper:

Yves Couder has provided experimental confirmation for classical single-particle interference. The greatest mystery of quantum mechanics has been solved by the good old classical field theory.

Let’s move to another “classical impossibility”: stable atoms with quantified energy levels. It is claimed that classical electrodynamics cannot account for stable atoms, and I would accept this claim, even if the situation is not as clear as it seems. For a possible counterexample I recommend this:

The author, Gryziński, has published its model in top peer-reviewed  magazines so I guess there is some truth about it.

But, for the sake of the argument, let’s assume that classical electrodynamics cannot explain the atom. Does it prove that no classical theory could do it? I don’t think so. We know that stable groups of charges with well-defined energies are possible. They are called ionic crystals. The charged particles do not move (if thermal motion is ignored), so there is no energy loss by radiation. The geometry of the crystal dictates the energy, which has a well-defined value. The reason for the stability of the crystals stays in the repulsive force generated by the orbiting electrons. One can argue that electrons are not composite particles, but a repulsive force can be provided by some modification of the electric field.

The third example I will discuss here is tunneling. Presumably, classical physics cannot explain this phenomenon because the particle does not have enough energy to overcome the barrier.

There are two observations I can make for this case:

a.       Neither the energy of the particle that tunnels, nor the value of the potential are accurately known at the moment of tunneling. They are average values, and these can be very different from the instantaneous ones.
b.      The actual force acting on the particle depends on the classical theory used to describe the experiment. A new theory could predict a much stronger force.

Objection 3

In the end I would like to point out a few reasons for investigating classical theories.

a.       If nature is not probabilistic after all, there is much to be discovered. Detailed mechanism behind quantum phenomena should be revealed, bringing out a deeper understanding of our universe, and maybe new physical effects.
b.      Quantum theories are not well equipped to describe the universe as a whole. There is no observer outside the universe, no measurement can be performed on it, not even in principle.
c.       Due to its inability to provide an objective description of reality, quantum mechanics may not be able to solve the cosmological constant problem. A theory that states clearly “what’s there” could provide a much better estimate of the vacuum energy. After all we are not interested in what energy someone could find by performing a measurement on the vacuum, but what the vacuum consists of, when no one is there to pump energy into it.
d.      Quantum mechanics requires an infinitely large instrument to measure a variable with infinite precision. When gravity is taken into account, it follows that local, perfectly defined properties cannot exist, because, beyond a certain mass, the instrument would collapse into a black hole.

For a detailed description of the latter two points, please read this paper by Nima Arkani-Hamed:

Thanks again, Florin, for this opportunity to present my arguments, and I am looking forward to seeing your opinion about them.

Andrei

## Heisenberg's matrix mechanics and the Uncertainty Principle

Continuing the exploration of various quantum mechanics formulations today I want to talk about Heisenberg's matrix mechanics. A major stumbling block for early 20th century physics was the explanation of discrete levels of energy exhibited for example by the atomic spectral lines. Classical mechanics was completely unable to provide any explanation and this crisis led to the exploration of new ideas.

One profound way of thinking about this was to strip the theory of all unobservable baggage (like the trajectory of an electron in an atom) and only talk about what can be actually tested by experiment. The discrete levels of energy would come out of the non-commutativity as follows:

1. Find n matrices $$Q_k$$ and $$P_k$$ such that $$[P_i, P_j] = [Q_i, Q_j] = 0$$ and $$[P_i, Q_j] = \frac{\hbar}{i}\delta_{ij}$$
2. Diagonalize the Hamiltonian $$H$$ by usage of the following transformation: $$S^{-1}Q_i S$$ and $$S^{-1}P_j S$$

The diagonal elements are the energy levels observed in experiments and the theory seems at first to have the same explanation power as the old semiclassical formulation of quantum mechanics due to Bohr and Sommerfeld, but lacking a mechanism to predict intensities. However this is not the case, because lurking about there is the standard Hilbert space formulation and von Neumann brought it to the surface in his classical book: The Mathematical Foundations of Quantum Mechanics. (Historical note: Heisenberg used intensity information and working backwards he derived a non-commutative rule of multiplication which turned out to be the usual matrix multiplication)

The diagonalization step is nothing but the usual eigenvalue-eigenvector problem: $$H x = \lambda x$$ where $$x$$ is a column of $$S$$ and the space of $$x$$ is the usual Hilbert space. As expected $$P, Q, H$$ are operators acting on this Hilbert space, and the intensities predictions arise out of projecting in the Hilbert space.

 Werner Heisenberg

Then as now, to defend a new paradigm one needs to produce a physical justification, and Mr. Heisenberg tried to explain the commutation relationship $$[P_i, Q_j] = \frac{\hbar}{i}\delta_{ij}$$ in terms of the inability to measure simultaneously the position and momenta of a particle in what is now known as the uncertainty principle. Today this is so commonplace that it is easy to overlook some subtle points about it and I'll present two such points.

Suppose that you have a beam of electrons moving in a straight line and with various speeds. By placing this beam in a magnetic field you can separate the electrons according to their speed. Then if you place a photographic plate in front of them you can also measure the electron's position. The momentum has an uncertainty, the position has its uncertainty as well, but if you compute $$\Delta p \Delta x$$ you seem to be able to beat the uncertainty principle!!! (try with the usual values of electron speed and detection resolution in an old cathode ray tube TV). However, this does not disprove the uncertainty principle and moreover Mr. Heisenberg was aware of this.

I can tell you the solution to this problem, but it is much more interesting for the reader to try to wrestle with this puzzle and explain the apparent contradiction (I'll provide Heisenberg's take on it in the next post).

The second subtle point (arising out of a advanced functional analysis) is that non-commutativity is not enough to establish the uncertainty principle. In addition one needs that:
1. the operators should be unbounded
2. the operators should either have no point spectrum or if they have a point spectrum that spectrum should lie outside the domain of the other operator.

Position and momenta respect those conditions, but spin does not and as such there is no uncertainty principle for spin! But wait a minute, this sounds preposterous. The commutator: $$[S_x, S_y] = i\hbar S_z$$ allow us to compute $$\Delta S_x \Delta S_y$$ and this is not zero! So how can we claim that there is no uncertainty principle for spin? There is a subtle difference between the spin and the position-momenta case. Can you spot it? Again' I'll present the solution next time because it is much more interesting for the reader to attempt to solve this connundrum as well.

To summarize, in the matrix mechanics formulation, both the spectra and the intensities arise out of the diagonalization step after the (infinite dimensional) matrices were selected to respect the commutation relationship. What the theory does not provide is a concrete transition mechanism (corresponding to the collapse postulate) and therefore quantum mechanics is intrinsically probabilistic. This left open the possibility to "complete" quantum mechanics and various interpretations of quantum mechanics are nothing but solution to the so-called "measurement problem". More on this in subsequent posts.

## Hamilton-Jacobi and Schrodinger equations

Quantum mechanics is a theory significantly larger than its standard textbook formulation. Quantum mechanics is categorical in nature and we need to go beyond parochial points of view and understand the links between different formulations and the categorical approach. Last time I presented a concrete problem in the phase space formulation, and today I want to arrive at the usual Hilbert space formalism by deriving Schrodinger's equation. There is no better way to do that than the follow the original paper by Schrodinger which I was pleasantly surprised to discover it is very clearly written and understandable.

The prerequisite of Schrodinger's approach is the Hamilton-Jacobi formalism and I will start with a short introduction of it.

The Hamiltonian equations of motion are:

$$\dot p = -\nabla_{x} H$$
$$\dot x = \nabla_{p} H$$

We want to explore changes of variables: $$(p,x) \rightarrow (P(p,x), X(p,x))$$ such that the Hamiltonian equations of motion remain valid in the new coordinates (those are called canonical transformations). This means that that differential form $$\omega = dp \wedge dx$$ is preserved:

$$dp \wedge dx = d(pdx) = d(PdX) = dP \wedge dX$$

which demands: $$d(pdx - P dX) = 0$$ or $$pdx - PdX = dR$$ So far so good, but we want to have a change of coordinates to simplify things. In particular we want the new Hamiltonian to be zero which will make the new $$X$$ and $$P$$ constants. So we have to see what will happen when we have time-dependent canonical transformations. By notation we say $$f_t(P,X) = f(P,X,t)$$. Then:

$$dS_t = pdx - P_t dX_t$$

Because
$$dR = \partial_t R dt + dR_t$$
$$dX = \partial_t X dt + dX_t$$

we get:

$$pdx - Hdt = PdX - (H + P\partial_t X + \partial_t R)dt + dR$$
which means that $$H+ P\partial_t X + \partial_t R$$ is the transformed Hamiltonian for the new coordinates: $$K(P(t), X(t))$$ The link between the new and old Hamiltonian is given by the use of a generating function $$S$$:

$$K = H + \partial_t S$$

where $$\nabla_x S = p$$ and  $$S = S(P,x,t)$$

If we demand $$K = 0$$ then $$P$$ and $$X$$ are constants. $$K = 0$$ is known as the Hamilton-Jacobi equation:

$$\partial_t S(x,t) + H(\nabla_x S(x,t), x) = 0$$

Because this is derived from the phase space Hamiltonian formalism, this equation is another formulation of classical mechanics in configuration space. So how does Mr. Schrodinger arrive at his equation which describes quantum mechanics?

 Erwin Schrödinger

The original paper can be found here and as I said it is remarkably accessible. The physical idea is that of the optical-mechanical analogy and the Eikonal equation in geometric optics because according to de Broglie the particles are waves (of an unspecified kind). In the short wavelength limit the waves becomes rays (and the particles are tracing a well defined path according to classical mechanics).

If we start from the Hamilton-Jacobi equation:

$$\frac{\partial S}{\partial t} + \frac{1}{2m}{(\frac{\partial S}{\partial x})}^2 + V = 0$$
$$p = \frac{\partial S}{\partial x}$$

we have

$$\frac{\partial S}{\partial t} = -H = -(T+V)$$
$$dS = \frac{\partial S}{\partial x}dx +\frac{\partial S}{\partial t } dt = p dx - H dt = L dt$$ - the Lagrangian and
$$S = \int_{t_0}^{t} L dt$$ - the action

With a usual decomposition: $$S = -Et + Q(x)$$:
$$\frac{\partial S}{\partial t} = -E$$

and from the H-J equation and our decomposition

$$|\nabla S| = \sqrt{2m(E-V)}$$ - which is the Eikonal equation.

Then Schrodinger considered surfaces of equal values for $$S$$ to see how this would propagate. The normal on those surfaces is:

$$dn = \frac{dS}{\sqrt{2m(E-V)}}$$

and the velocity:

$$u = \frac{E}{\sqrt{2m(E-V)}}$$

Those surfaces almost respect a wave equation but to take de Broglie's idea seriously Schrodinger forced the solution to be a wave:

$$\psi = A e^{\frac{i S}{\hbar}} = A e^{\frac{-iEt}{\hbar} + \frac{iQ(x)}{\hbar}}$$

respecting a wave equation according to the geometric optics insight (thus modifying the Hamilton-Jacobi equation in the process):

$$\Delta \psi -\frac{\ddot \psi}{u^2} = 0$$

Because $$u^2 = \frac{E^2}{2m(E-V)}$$, Schrodinger's equation is:

$$(-\frac{\hbar^2}{2m} \Delta + V )\psi = E \psi$$

It is informative to see what we get if we directly plug the ansatz $$\psi = A e^{\frac{i S}{\hbar}}$$ into the Hamilton-Jacobi equation. Simple calculations yields:

$$(-\frac{\hbar^2}{2m \psi} \Delta + V )\psi = E \psi$$

Conversely, if we plug the ansatz into the Schrodinger equation we recover the Hamilton-Jacobi equation and an additional term proportional with $$\hbar$$.

Schrodinger's equation and the Hamilton-Jacobi equations are cousins and they both are equations in the configuration space.

There are typical mistakes people make about them. For example working with a single particle, one gets the temptation to identify the particle with its wavefunction, This mistake is cured when considering the wavefunction of N particles where such identification is impossible because we are working in configuration space. Other mistake is "deriving" Schrodinger equation directly from Hamilton-Jacobi, like in this paper: http://arxiv.org/pdf/1204.0653v1.pdf. If that were true, quantum mechanics would be nothing but classical mechanics. In this paper the mistake occurs in Eq. 4.5 because total and partial derivatives cannot be interchanged. Here is a simple counterexample: $$x(t) = e^t$$

$$\frac{\partial \frac{d x}{d t}}{\partial x} = \frac{\partial x}{\partial x} = 1$$ but
$$\frac{d \frac{\partial x}{\partial x}}{dt} = \frac{d 1}{dt} = 0$$

The original triumph of Schrodinger's equation (derived from free particle considerations) was its application to the bound states of the atom. There the matrix mechanics of Heisenberg was developed to eliminate unobservable classical concepts like the classical path and later on Schrodinger proved the equivalency of the two seemingly different approaches.

But what is the link between Schrodinger's equation and the categorical formulation of quantum mechanics using the abstract products $$\alpha$$ and $$\sigma$$? First we need to pass from the Schrodinger picture where the wavefunction evolves in time and the operators are stationary (recall that in Hamilton-Jacobi $$\dot P = \dot Q = 0$$) to the Heisenberg picture where the states are stationary and the operators change in time. Here the time evolution of the operators is given by the abstract product $$\alpha$$:

$$\dot A = H\alpha A$$

The categorical formulation of quantum mechanics is a formulation in a Hilbert (and phase space when possible) in the Heisenberg picture.

## Quantum harmonic oscillator in phase space

Continuing the physics posts, today I want to show how to solve a standard quantum mechanics problem: the harmonic oscillator, but do it in the unusual phase space formulation of quantum mechanics. It is not until we can solve concrete problems that we truly understand a new formalism. For today's result I will follow a pedagogical paper by Emile Grgin and Guido Sandri from 1996.

The Hamiltonian of the harmonic oscillator is obviously:

$$H=\frac{1}{2m} p^2 + \frac{m\omega^2}{2} x^2$$

and it is advantageous to work with dimensionless variables as follows:

$$p=\sqrt{\hbar \omega m} \eta$$
$$x = \sqrt{\hbar / \omega m} \xi$$
$$H = \hbar \omega \chi$$

If we recall the abstract products $$\alpha$$ and $$\sigma$$ in phase space:

$$\alpha = \frac{2}{\hbar} sin(\frac{\hbar}{2} \overleftrightarrow{\nabla})$$
$$\sigma = cos(\frac{\hbar}{2} \overleftrightarrow{\nabla}))$$

where

$$\overleftrightarrow{\nabla} = \frac{\overleftarrow{\partial}}{\partial \eta}\frac{\overrightarrow{\partial}}{\partial \xi} - \frac{\overleftarrow{\partial}}{\partial \xi}\frac{\overrightarrow{\partial}}{\partial \eta}$$
$$f\overleftrightarrow{\nabla}g = \{f,g\}$$: the Poisson bracket

with $$\eta$$ and $$\xi$$ cannonical variables (dimensionless momenta and position) and the funny arrows are used as a reminder of how they act: on the left or on the right arguments.

In phase space if $$u$$ is a state, the condition for a pure state is:

$$u=u\sigma u$$

the expectation value for an observable $$f$$ and a state $$u$$ is:

$$\langle f {\rangle}_{u} = \int f \sigma u ~dx dp$$

and the characteristic equation corresponding to the eigenvalue-eigenvector problem in standard Hilbert space formalism is:

$$f \sigma u_{\lambda} = \lambda u_{\lambda}$$

In the Hilbert space formulation, the time-independent Schrodinger equation in the dimensionless parameters reads:

$$\frac{1}{2}(\xi^2 - {\partial_{\xi}}^2) \psi = \lambda \psi$$

and the solution is well known:

$${|\psi|}^2 = \frac{1}{\sqrt{\pi}2^n n!} H_n^2 (\xi) e^{-\xi^2}$$
$$\lambda = n+1/2$$
with $$H$$ the Hermite polynomials.

Now what do we get in the phase space formulation? The Hamiltonian is:

$$\chi = \frac{1}{2}(\xi^2 + \eta^2)$$

and we can compute $$\chi \alpha$$ and $$\chi \sigma$$:

$$\chi \alpha = \eta \partial_{\xi} - \xi \partial_{\eta}$$
$$\chi \sigma = \frac{1}{2}(\xi^2 - \hbar^2 / 4 \partial^2_{\xi}) + \frac{1}{2}(\eta^2 - \hbar^2 / 4 \partial^2_{\eta})$$

because higher powers of $$\overleftrightarrow{\nabla}$$ vanish as they apply on the left side as well.

$$\chi \alpha$$ is the same as in classical mechanics and $$\chi \sigma$$ is a double copy of the Hamiltonian in the Hilbert space formalism!!! And this is the key to prove the equivalence between the two approaches. What you get from the characteristic equation is a sum of the products of the Hermite polynomials which by some Hermite polynomials identity magic reduces to the desired result. But those are mathematical details relevant only to test one's knowledge of functional analysis. The full mathematical details can be found here and here. We are after the conceptual ideas.

For the problem above the link between the phase space state $$u$$ and the Hilbert state $$\psi$$ is given by:

$${|\psi(\xi)|}^2 = \frac{1}{2\pi}\int_{-\infty}^{+\infty} u(\xi, \eta) ~d\eta$$

What I presented so far is the quantum mechanics formulation in phase space using the observable product $$\sigma$$. But the products $$\sigma$$ and $$\alpha$$ can be combined to create an associative product $$\star = \sigma + J \frac{\hbar}{2}\alpha$$ known as the Moyal, or the star product.

 Jose Enrique Moyal
Recalling that $$u\sigma u = u$$, similarly in the star product formulation the pure states $$F$$ respect:

$$(2 \pi \hbar) F\star F = F$$

and the eigenvalue-eigenvector problem is not surprisingly:

$$H\star F = \lambda F$$

Now we can combine all this and past posts to introduce the high level overview of quantum mechanics in phase space and its relationship with quantum mechanics in Hilbert space.

Phase space:

observables = differentiable functions on phase space
generators = vector fields on phase space
1-to-1 map observable and generators: J with $$J^2 = -1$$

observable product: $$\sigma = cos(\frac{\hbar}{2} \overleftrightarrow{\nabla})$$
generator product: $$\alpha = \frac{2}{\hbar} sin(\frac{\hbar}{2} \overleftrightarrow{\nabla})$$
associated product: $$\star = e^{J\frac{\hbar}{2}\overleftrightarrow{\nabla}}$$
state space (for the products $$\sigma$$ or $$\star$$) = phase space

Hilbert space:

observables = hermitean operators
generators = anti-hermitean operators
1-to-1 map observable and generators: J with $$J^2 = -1$$

observable product: $$A\sigma B = \frac{1}{2}(AB + BA)$$ - the Jordan product
generator product: $$A\alpha B = \frac{J}{\hbar}(AB-BA) = \frac{J}{\hbar} [A,B]$$
associated product: $$A\cdot B = A B$$ regular operator multiplication
state space (for the products $$\sigma$$ or $$\cdot$$) = Hilbert space

The states can be  defined for the observables product or for the associated product. In the first case one encounters the Jordan-GNS construction and in the second case the usual GNS construction.

One goes from the phase space formalism to the Hilbert space formalism by a quantization procedure, and the simplest one is Weyl quantization which directly constructs the operators in the Hilbert space. Many other quantization procedures are known.

The same thing is valid for classical mechanics with the main difference that $$J^2 = 0$$!!!! There the product $$\alpha$$ is the Poisson bracket, and the product $$\sigma$$ is regular function multiplication on phase space.

Deformation quantization is the process which transforms the products $$\alpha$$ and $$\sigma$$ from classical to quantum realizations. the dimension of the map $$J$$ is $$\hbar$$ and this observable to generator map (known as dynamic correspondence) corresponds to Noether's theorem.

So now what? How can we understand quantum mechanics? There are basically two camps, the ontic and the epistemic ones. If you are in the ontic camp you you may like the Bohmian interpretation where particles have a well define position at all times. But what prevents us to attach an ontic interpretation to the phase space formulation as well? After all phase space is well understood and classical mechanics does not have any interpretation issues. Sure, in the phase space formulation one encounters quasi-probabilities in the form of Wigner functions, but you have to compare their strangeness this with the strangeness of the quantum potential. Something has to be different than in the case of classical mechanics otherwise you don't get quantum effects.

If we attach an ontic meaning to quantum mechanics in phase space, now we have two distinct ontic interpretations. But an ontic interpretation must be unique, otherwise it cannot be taken seriously.

The existence of the phase space formulation of quantum mechanics presents the greatest challenge to the Bohmian interpretation. This formalism has the same limitations as Bohmian quantum mechanics particularly in the treatment of spin which is a pure Hilbert space phenomena with no classical counterpart. Why should non-detectable violations of relativity (in the case of Bohmian quantum mechanics) be better than non-detectable violations of positive probabilities (in the case of quantum mechanics in phase space)?

## Is China's turmoil the next Lehman Brothers?

I am taking a break from physics topics to discuss a hot topic of the day: the extreme turmoil on the stock markets due to trouble in China. I am not sure how the system works in other countries, but in US your retirement funds, your nest egg, is usually tied to the stock market fortunes as the largest investors are mutual funds. Every person have his or her tolerance to fluctuations and loss but major evaporation of retirement funds can be very nerve racking. Since I got burned in the past and learned from my mistakes, I want to share some common sense advice you will not easily find otherwise.

The first observation is that unless you are a serious investor willing to commit time and resources into researching individual stocks it makes little difference overall on which mutual fund you select. Each mutual fund is already diversified and by watching how the DOW does you have a pretty good idea how your investments are doing. For the usual ups and downs of the market you do not need to worry at all, you will have an average return of about 8%, but what to do when there are violent price movements like the recent ones?

If you watch the DOW value, get in the habit to see watch the transaction volume as well. The idea is that large financial institutions do have very serious experts who make decent predictions, and more important, those advice is actually followed. And when large mutual funds decide to make a large move, the volume tells the story. So the advice is simple:
• when DOW drops significantly but the volume is close to the average, don't panic and do nothing.
• when DOW drops significantly and the volume is two to three times the average it is time to sell. The large players decided that the outlook is negative.
• when DOW increases significantly but the volume is close to the average, do nothing.
• when DOW increases significantly and the volume is two to three times the average it is time to buy. The large players decided that the outlook is positive.
Sure, you will miss the first big drop, and the first large increase, but in the long term this does not matter. This advice is for the investors who don't have the stomach for large drops in value. If your tolerance for risk is higher or you have the luxury of a really long term investment period measured in many decades then you should not care about the stock market antics in the first place.

Now on the current China turmoil.

Is this a short storm, or the harbinger of a larger trouble like the Lehman Brothers collapse? Honestly, I don't know, and I think nobody knows either. China is now the second largest economy and any turmoil there has large implications. Also the current trouble is the result of irresponsible advice from Chinese leadership to ordinary citizens to buy stocks in a get rich quick scheme. This created a bubble which now burst. But what will the effect be in the psychology of ordinary Chinese citizens? Would this create unbearable social pressure which will result in the change of the political system? Would this impact the real Chinese economy? Nobody knows because this is a first in the modern history of China. When Lehman collapsed, for a few months there was no economic pain. Everyone was reassured that "the economic fundamentals are strong" and this was just a blimp on the radar. But something funny happened: the usual economic activity simply stopped like a switch was turned off because lending between banks froze solid due to sudden lack of confidence. It took a Keynes style influx of money to get the economic engine restarted.

## Where does the Hilbert space come from?

Continuing the discussion from last time, today we can put some of the pieces of the puzzle together. It is helpful to switch the discussion direction from classical to quantum and start for a moment from the quantum mechanics side to see where we want to arrive.

Why do we use a Hilbert space in quantum mechanics? This is a big topic and we cannot cover it in one (or even several posts). Right away we will restrict ourselves to the finite dimensional degrees of freedom case, thus excluding the field theory considerations and avoiding the issues raised by Haag's theorem, or the breakdown of the Stone-von Neumann uniqueness theorem. We will also skip the treatment of unbounded operators which require the theory of rigged Hilbert spaces and we will stick with boring but well behaved bounded operators.

For a bounded operator T on a Hilbert space it is easy to prove that $$||T^{\dagger} T|| = {||T||}^2$$ as follows:

$${||T \Phi||}^2 = \langle T \Phi, T \Phi\rangle = \langle T^{\dagger} T \Phi, \Phi\rangle \leq ||T^{\dagger} T\Phi || ||\Phi|| \leq ||T^{\dagger} T || {||\Phi||}^2$$
therefore
$${||T||}^2 \leq ||T^{\dagger} T ||$$
and since
$$||T^{\dagger} T || \leq ||T^{\dagger}|| ||T|| = {||T||}^2$$
we have:
$${||T||}^2 \leq ||T^{\dagger} T || \leq {||T||}^2$$

An algebra of bounded operators on a Hilbert space is the prototypical example of a C* algebra. A remarkable fact is the correspondence between states and representations of C* algebra given by the GNS construction. Here a representation is a linear map from the elements of the C* algebra to bounded operators on some Hilbert space.

From categorical considerations one can obtain a C* algebra without the norm axioms. To distinguish math from physics one needs to be able to make experimental predictions and this is where the states enter the picture. A state on a C* algebra gives rise to a representation of the algebra as bounded linear operators on some Hilbert space and this is how Hilbert spaces are introduced. The key ingredient for this to work is the C* norm condition: $$||T^{\dagger} T|| = {||T||}^2$$. However, this norm is unique and is given by the spectral radius - an algebraic concept! So there is hope we can arrive at quantum mechanics using only algebraic methods. Now we will show how.

Coming back to the quantization discussion from the prior post, what we need to achieve is a prescription which constructs operators on a Hilbert space from functions on the phase space (also known as the cotangent bundle). Even better we should be able to start from either a Kahler, symplectic, or Poisson manifold.

We can start with the simplest case where we replace the position $$q$$ and the momenta $$p$$ with the operators: $$x$$ and $$\frac{h}{i}\frac{\partial}{\partial x}$$ in any observable $$f(p, q)$$ provided $$f$$ contains no products $$pq$$ because the position and momenta operators in the Hilbert space do not commute and the order of the operators is ambiguous.

The next level of sophistication is Weyl quantization procedure and the details can be found here. (Please excuse me for skipping typesetting it in LaTeX.) Weyl quantization tends to preserve well symmetry properties, but a better quantization prescription is Berezin quantization which work on all Kahler manifolds when positivity is guaranteed by the Kodaira embedding theorem.

 Erich Kahler

A Kahler manifold is a truly outstanding mathematical object where three concepts meet:

• a metric structure
• a symplectic structure
• a complex structure
and any two define the third one. The main example is the complex projective space (endowed with the Fubini-Study metric) which is essential for quantum mechanics. It is very enlightening to see how it all works out in quantum mechanics and I'll attempt to show it below.

In classical and quantum mechanics there are two products, one symmetric $$\sigma$$ and one anti-symmetric $$\alpha$$ corresponding to observables and generators as follows:

Observables: $$\sigma$$ = regular function multiplication on phase space OR Jordan product
Generators: $$\alpha$$ = Poisson bracket OR commutator

There is also a 1-to-1 map $$J$$ between observable and generators called dynamic correspondence where $$J^2 = 0$$ for classical mechanics and $$J^2 = -1$$ for quantum mechanics. This map corresponds to Noether's theorem.

Composing two physical systems 1 and 2 gives rise to the following fundamental composition relationship:

$$\sigma_{12} = \sigma_1 \otimes \sigma_2 + J^2 \frac{\hbar^2}{4}\alpha_1 \otimes \alpha_2$$
$$\alpha_{12} = \sigma_1 \otimes \alpha_2 + \alpha_1 \otimes \sigma_2$$

and so the symmetry and anti-symmetry of the products is preserved.

Now we want to deform the Poisson bracket and regular function multiplication of classical mechanics which respects the composition with $$J^2 = 0$$ into two products which respect $$J^2 = -1$$. We can do this term by term in powers of $$\hbar$$ preserving associativity at each step. This is the essence of deformation quantization.

Without ado, here is the solution given by Moyal sine and cosine brackets in terms of the Poisson bracket $$\{ , \}$$ in the simplest case of a flat space:

$$\alpha = \frac{2}{\hbar} sin (\frac{\hbar}{2} \{ , \})$$
$$\sigma = cos (\frac{\hbar}{2} \{ , \})$$

The star product is then $$\star= \sigma+ J\frac{\hbar}{2} \alpha$$ and we arrived at quantum mechanics in phase space.

First a note: I demanded earlier to preserve associativity at each power of $$\hbar$$. This is a physical requirement to be able to compose experiments sequentially and not care where we draw the boundaries between them. But this has a very interesting consequence: we have freedom of pick how we carry out the quantization at each power of $$\hbar$$ step and this makes the subject of quantization non-trivial. In particular it turns out that the equivalence classes of star products on symplectic manifolds are in 1-to-1 correspondence with the second de Rham cohomology $$H^2_{dR} (M)$$!

Second, we can see where the inner product is coming from. From the Moyal sine bracket we extract a symplectic form $$\omega^{IJ}$$ and construct it's inverse $$\Omega_{IJ}$$. So we have one of the three structures of a Kahler manifold: the symplectic structure. But we also have the complex structure as well because we have $$J^2 = -1$$. It can be shown that $$J$$ is actually a tensor or rank (1,1): $$J = J^{I }_{ J}$$ and from this we get a metric tensor $$g_{IJ}$$:

$$g_{IJ} = \Omega_{IK} J^{K}_{ J}$$

The complex inner product is defined now by: $$g+ \sqrt{-1}\Omega$$:

$$\langle X, Y \rangle = X^{T} g Y + i X^T \Omega Y$$

where X and Y are column vectors : $$q_1, q_2,...q_n, p_1, p_2, ..., p_n$$

Time evolution preserves $$J$$ and $$\Omega$$, meaning they preserve the metric structure by preserving a normalization constraint:

$$\langle g \rangle - 1= X^I g_{IJ} X^J - 1 = 0$$

The constraint Hamiltonian motion which preserves the metric structure is nothing but the Schrodinger equation is disguise!

I do not want to create the impression that this is all as simple as this. I only discussed the flat $$R^{2n}$$ case above. There are many subtle and hard problems, as well as open questions. As an example, there are Poisson manifolds which do not admit a Kahler structure, but all Poisson manifolds are quantizable. How would the quantization of such a system look like? Perhaps there are no bounded operators in this case, I don't know.

Next time I'll present a concrete calculation of a standard problem in the phase space formalism of quantum mechanics. This will challenge the ontic interpretation of quantum mechanics.