52 votes
Accepted

How can Planck's constant take different values?

In a purely classical (Newtonian) universe, quantum effects would be absent, and the way to pretend this is true mathematically is to allow Planck's constant to approach zero, and see what the ...
niels nielsen's user avatar
28 votes
Accepted

On Groenewold's Theorem and Classical and Quantum Hamiltonians

You probably need to internalize Ivan Todorov's accessible Quantization is a mystery. Your best bet for addressing your questions is Geometric quantization, not phase space quantization that you ...
Cosmas Zachos's user avatar
17 votes

Is Planck's Constant Really a Constant?

How did he assume that $\frac{\hbar}{i}=1$? He didn't. Check the definition he gives of the commutator in equation (1.02). And if $\hbar$ (as we have learned it) is a constant how can we say that ...
TheoPhy's user avatar
  • 900
16 votes

Is Planck's Constant Really a Constant?

Groenewold is working in the framework of deformation quantization, where the (reduced) Planck constant $\hbar$ is treated as a formal parameter that doesn't have to be the actual physical value $\sim ...
Qmechanic's user avatar
  • 203k
12 votes
Accepted

Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?

"Used in anger" or "killer ap"? To my knowledge, no problem has been solved in the phase-space quantization language that was not solvable in the other two formulations/pictures (Hilbert space or path ...
Cosmas Zachos's user avatar
12 votes
Accepted

Dequantizing Dirac's quantization rule

I do not know about deep questions. And people seem to give pretty deep answers here. My contribution is to show $$ \lim_{\hbar \to \infty} \frac{1}{i\hbar}[ F(p,x) , G(p,x)] = \{F(p,x), G(p,x)\}_{P....
OkThen's user avatar
  • 824
12 votes

Dequantizing Dirac's quantization rule

Let me rearrange the logic of the Moyal Bracket that @ACuriousMind discussed neatly, by visiting a notional planet where people somehow discovered classical mechanics and quantum mechanics ...
Cosmas Zachos's user avatar
9 votes

What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

The so-called (generic) failure of the quantum Liouville theorem, i.e. the (generic) violation of the continuity equation $$ \rho~{\rm div}_{\rho} X^Q_{-H} + \frac{\partial \rho}{\partial t}~\neq~0 \...
Qmechanic's user avatar
  • 203k
8 votes

What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?

Apologies for my inability to share intuition, a frequently subjective issue... I have learned a lot by reading the Steuernagel group numerical flows and topological features of such flows, in ...
Cosmas Zachos's user avatar
8 votes

How can Planck's constant take different values?

Theories that are said to have real parameters really have a multidimensional parameter space, and the real parameters are coordinates in that space. Often, multiple points in the parameter space ...
benrg's user avatar
  • 26.2k
7 votes

Dequantizing Dirac's quantization rule

The statement is true by the very definition of quantization, i.e. there is nothing to show. So let's talk about the definition of quantization, which is a map from classical observables to quantum ...
ACuriousMind's user avatar
  • 125k
6 votes

Is Planck's Constant Really a Constant?

For a slightly different perspective, in natural units one can set $\hbar = 1$. That is, in natural units we agree to measure action in units of $\hbar$ (instead of, say, $\rm J\cdot s$). Seen this ...
Charles Hudgins's user avatar
5 votes
Accepted

Solving the *-genvalue equation of a free particle

Basically there is, for the real part of the Wigner function, Lemma 3 (pp 27-29) of my book, CTQMPS. This is the basic exercise any student of that structure should do, whether instructed to, or not. ...
Cosmas Zachos's user avatar
4 votes
Accepted

Proof of "non-existence" of marginals of the Husimi $Q$-function

The stuff is all in Husimi's original 1940 paper, but you have to work at it... So I'll just illustrate the point based on our concise summary of the material in Wolfgang's outstanding treatise, eqns (...
Cosmas Zachos's user avatar
4 votes

Classical limit in deformation quantization

The following comments seem relevant to OP's post: In deformation quantization an associative star product $$\star:~~C^{\infty}(M)[[\hbar]]\times C^{\infty}(M)[[\hbar]]\to C^{\infty}(M)[[\hbar]] \...
Qmechanic's user avatar
  • 203k
4 votes
Accepted

Derivation question of WKB method

In the semiclassical WKB approximation the Planck constant $\hbar$ is not a fixed number equal to its physical value $\approx 1.05 \times 10^{-34} Js$. Instead it is an indeterminate. The ...
Qmechanic's user avatar
  • 203k
4 votes
Accepted

Symplectic reduction to moduli space in Chern-Simons theory

You can write the symplectic form on the large phase space as $$\Omega(\delta_i A,\delta_j A) = \frac{k}{4\pi} \int_\Sigma \langle\delta_i A \wedge \delta_j A\rangle.$$ Here $\langle, \rangle$ is ...
Ryan Thorngren's user avatar
4 votes

Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?

The Moyal-Liouville equation is widely used in condensed matter problems, where it is at the heart of the so-called transport formalism, or kinetic theory. It is also widely used in quantum optics. In ...
FraSchelle's user avatar
  • 10.5k
4 votes
Accepted

Non-commutative Fourier transform of an operator

There are very compelling, and in my opinion enlightening reasons to call the Weyl transform a noncommutative Fourier transform. Background Classical theories can be seen as (classical) probability ...
yuggib's user avatar
  • 12k
4 votes
Accepted

What is a Borel subalgebra?

I'll give you here a physics motivated definition of Borel subalgebras. I'll start with the case of $SL(2)$, which is the case of interest when treating quantum $SU(2)$, but also generalize the ...
David Bar Moshe's user avatar
4 votes
Accepted

Operator traces in Kontsevich quantization

Wikipedia says the following properties to uniquely determine the trace operation (up to scalar multiples): $\mathrm{tr}(cA) = c\mathrm{tr}(A)$ $\mathrm{tr}(A + B) = \mathrm{tr}(A) + \mathrm{tr}(B)$ $...
Daniel's user avatar
  • 1,184
4 votes

A Hamiltonian with a potential depending on the momentum

You have an embarrassment of riches, and you have to use symmetry or other physical information to restrict your choice! This is dubbed the "operator ordering ambiguity problem", and has ...
Cosmas Zachos's user avatar
4 votes
Accepted

Reconciling the expression for the Wigner function involving $\langle x+\xi/2|\rho|x-\xi/2\rangle$ with the one using the characteristic function

You have jammed up your shift variables, making them do double duty; and, importantly, you have used different conventions for $\hbar$: (1) requires that it be equal to 1, while (2) to 2. In any case,...
Cosmas Zachos's user avatar
4 votes

How can Planck's constant take different values?

From a mathematical perspective, it is interesting to study quantum theories, where the (reduced) Planck's constant is not necessarily equal to its physical value. For instance, In the mathematical ...
Qmechanic's user avatar
  • 203k
3 votes

Physical interpretation of differences between classical and quantum ensemble dynamics

Cosmas Zachos has already given a nice answer. He correctly points out that the sine function in the $\star$-commutator originates from the exponential function in the $\star$-product. Question: ...
Qmechanic's user avatar
  • 203k
3 votes

Heisenberg group deformation

I'm not sure I am understanding the quest, but, in plain phase space (q,p) with the standard Groenewold(-Moyal) star product, and arbitrary deformation parameter, here taken to be i, your construct is ...
Cosmas Zachos's user avatar
3 votes

Quantum systems without a classical analogue?

1) It is true that not all quantum systems have classical analogues. E.g. if we have a quantum algebra $({\cal A},\ast)$ of Laurent polynomials in an indeterminate $\hbar$, and endowed with an ...
Qmechanic's user avatar
  • 203k
3 votes

Angular Momentum Addition in Phase Space QM

I could give you an answer by barking up a very different tree indeed! In phase space QM, and not, repeat not geometric quantization, you may work on flat phase spaces and forfeit spheres altogether, ...
Cosmas Zachos's user avatar
3 votes

Weyl Ordering Rule

Another way to look at this: $e^{ix\hat{P}+ik\hat{Q}}$ is automatically Weyl-ordered. This is because each term in the Taylor expansion, $\frac{1}{n!}(ix\hat{P}+ik\hat{Q})^n$, is Weyl-ordered. You can ...
Technically Natural's user avatar
3 votes
Accepted

Classical limit in deformation quantization

It might be worth reviewing formulas such as (122,34,5; 131) of our book, but, frankly, I am not sure I understand what underlies and follows the question, and especially the classical limit ...
Cosmas Zachos's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible