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 ...
- 97.8k
31
votes
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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 ...
- 66.3k
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 ...
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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 ...
- 213k
12
votes
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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 ...
- 66.3k
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
$$
\lim_{\hbar \to \infty} \frac{1}{i\hbar}[ F(p,x) , G(p,x)] = \{F(p,x), G(p,x)\}_{P....
- 824
12
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Dequantizing Dirac's quantization rule
Let me rearrange the logic of the Moyal Bracket that @ACuriousMind discussed neatly, by visiting a notional planet where people somehow discovered classical mechanics and quantum mechanics ...
- 66.3k
9
votes
What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics?
The so-called (generic) failure of the quantum Liouville theorem, i.e. the (generic) violation of the continuity equation
$$ \rho~{\rm div}_{\rho} X^Q_{-H} + \frac{\partial \rho}{\partial t}~\neq~0 \...
- 213k
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 ...
- 66.3k
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 ...
- 28.7k
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 ...
- 129k
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 ...
- 1,590
5
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]] \...
- 213k
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.
...
- 66.3k
5
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 ...
- 31.2k
5
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 ...
- 66.3k
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 (...
- 66.3k
4
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 ...
- 66.3k
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 ...
- 213k
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 ...
- 12.2k
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 ...
- 8,144
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 ...
- 10.8k
4
votes
Accepted
Operator traces in Kontsevich quantization
Wikipedia says the following properties to uniquely determine the trace operation (up to scalar multiples):
$\mathrm{tr}(cA) = c\mathrm{tr}(A)$
$\mathrm{tr}(A + B) = \mathrm{tr}(A) + \mathrm{tr}(B)$
$...
- 1,184
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,...
- 66.3k
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 ...
- 213k
4
votes
Classical limit of Moyal bracket in integral representation
There are no compact integral forms of the PBs, of course, unless you consider conversions of derivatives into powers of the Fourier conjugate variable, as you might be insinuating in the comments. ...
- 66.3k
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: ...
- 213k
3
votes
Derivation question of WKB method
I'm personally quite averse to treating $\hbar$ as "small parameter," since it has dimension and can be made arbitrarily big or small by suitable choice of units. Whenever this slight of ...
- 993
3
votes
How to derive the formula for the quantum Hamiltonian $\hat H(\hat P,\hat Q)$ in terms of the classical $H(q,p)$, via Weyl Ordering?
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 ...
- 4,019
3
votes
Accepted
Groenewold's derivation of the star product (On the principles of elementary Quantum Mechanics)
One can see it directly, but Cosmas Zachos' suggestion "to working in multi-Fourier space so there are no derivatives present" is the most systematic & fool-proof approach. Here is a sketched ...
- 213k
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