31

Indeed, canonical quantization works just when it works. It is in my view wrong and dangerous to think that this is the way to construct quantum theories even if it sometimes works: it produced astonishing results as the theoretical explanation of the hydrogen spectrum. However, after all the world is quantum and classical physics is an approximation: the ...


29

Be aware that there exist various definitions of a canonical transformation (CT) in the literature: Firstly, Refs. 1 and 2 define a CT as a transformation$^1$ $$ (q^i,p_i)~~\mapsto~~ \left(Q^i(q,p,t),P_i(q,p,t)\right)\tag{1}$$ [together with a choice of a Hamiltonian $H(q,p,t)$ and a Kamiltonian $K(Q,P,t)$; and where $t$ is the time parameter] that ...


24

I understand the definition of a Hilbert space. But I do not understand why non-commutativity compels us to use Hilbert spaces. It doesn't, but that's not what Scrinzi is saying. The reason is doesn't is because we could work, for example, in Wigner quasiprobability representation: $$\rho\mapsto W(x,p) = \frac{1}{\pi\hbar}\int_{-\infty}^\infty\langle x+x'|\...


21

It's subtle. The theorem is not there: quantum flows are compressible (Moyal, 1949). I'll follow Ch. 0.12 of our book, Concise Treatise of Quantum Mechanics in Phase Space, 2014. The analog of the Liouville density of classical mechanics is the Wigner function in phase space quantum mechanics. Its evolution equation (generalizing Liouville's) is $$ {\...


20

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 appear to be hung up on. Since I did not follow the logic of your conclusion/question after the Dirac 1925 thesis heuristic vision map you wrote, and I have no broad ...


19

Let me actually collect most of my comments in an answer attempting to be more coherent than they, or your labile question. In fact, you are piling up three different questions, logically distinct, but with strong and natural connections, so it might be worth splitting them apart, before bringing them back together in the final coda. First, there is plain ...


18

Because you do equilibrium statistical mechanics. In the usual ensemble theory we associate to a system (a macrostate) a big number of corresponding microstates, each microstate is a point in phase space, that point is called representative point. Now you want to study how these points move in phase space. First of all the situation of equilibrium for the ...


17

Your interpretation is not quite right. One sharp interpretation one can give to this "cutting" of phase space into cubes of size $h^{2N}$ (here $N$ is the dimension of the system's configuration space), is that it allows one to use classical phase space to count the number of energy eigenstates of the corresponding quantum hamiltonian. Instead of trying ...


17

So, the short answer is that you're quite correct: if the dynamics of a system is subject to Liouville's theorem, then phase space volume is conserved, so the entropy associated to a given probability distribution remains constant as it evolves under those dynamics. This is actually just one instance of a much more general puzzle: how do we reconcile the ...


15

Explicitly proving non-integrability of an arbitrary Hamiltonian system is an open problem. For some classes of Hamiltonian systems (e.g systems on a plane) is possible to prove explicitly the non-integrability of the system, using theorems of Poincare, Burns, Ziglin and Yoshida (and generalizations). For example there is a theorem of Poincare: For a ...


15

There is no "wavefunction in phase space" because the wavefunctions $\psi(x)$ and $\psi(p)$ are obtained from the abstract state vector $\lvert \psi\rangle$ by $\langle x\vert \psi\rangle$ and $\langle p\vert \psi\rangle$, respectively. Since position and momentum don't commute, there are no $\lvert x,p\rangle$ to get a naive $\psi(x,p)$. However, from any ...


14

Cool question! Thanks to user lionelbrits for his answer that prompted me to pull out my mechanics books and check the definitions of "canonical transformation" given by different authors. If you look in Goldstein's classical mechanics texts in the section on canonical transformations, then you'll find that canonical transformations are essentially defined ...


14

In brief: phase space is not made into a vector space because that additional structure provides no benefit; quantum mechanics uses a Hilbert space because that additional structure does provide benefits. Any time you relate a mathematical structure to a physical concept, you need to ask how useful that relation is. The mathematical structure will have ...


13

Consider a non-relativistic massless particle with charge $q$ on a 2D torus $$\tag{1} x ~\sim~ x + L_x , \qquad y ~\sim~ y + L_y, $$ in a constant non-zero magnetic field $B$ along the $z$-axis. Locally, we can choose a magnetic vector potential $$\tag{2} A_x ~=~ \partial_x\Lambda, \qquad A_y ~=~ Bx +\partial_y\Lambda, $$ where $\Lambda(x,y)$ is ...


13

The importance comes from the equations of motions: $$ \dot{q}=\frac{\partial H}{\partial p}\, ,\qquad \dot{p}=-\frac{\partial H}{\partial q} $$ which can be rewritten as $$ \dot{q}=\{q,H\}\, ,\qquad \dot{p}=\{p,H\} $$ In particular, for an arbitrary function of $f(p,q)$, we have $$ \frac{d}{dt}f(p,q,t)=\{f,H\}+\frac{\partial f}{\partial t}\, . $$ ...


13

Yes. A phase space trajectory of a smooth system$^1$ has to be a continuous curve. For it to be called "periodic", the movement has to repeat itself, both velocity and position: i.e., it must come back to the same spot in phase space. For a deterministic system, the current condition determines uniquely its future evolution, and there can not be "...


13

In classical physics, the definition of an observable is that it is a (reasonably smooth) function of position and momentum. Therefore, quantum systems obtained purely by quantization of a classical system also have that all their observables are functions of position and momentum. In quantum physics, an observable is just an operator designate to belong to ...


13

Wrong? There is nothing "wrong". The phase-space formulation of QM, for which the Wigner-Weyl ordering prescription is a particularly popular flavor, is, indeed, completely equivalent to the Hilbert-space, and also the path-integral formulations. Equivalence of results does not dictate equivalence of ease, techniques, and convenience. Ask the same ...


12

Hints to the question (v1): We cannot resist the temptation to generalize the background spacetime metric from the Minkowski metric $\eta_{\mu\nu}$ to a general curved spacetime metric $g_{\mu\nu}(x)$. We use the sign convention $(-,+,+,+)$. Let us parametrize the point particle by an arbitrary world-line parameter $\tau$ (which does not have to be the ...


12

For a general and brief overview of the mathematical framework of Quantum Mechanics, see this answer. In a nutshell, Hilbert spaces arise from the representation theory of C*-algebras, which are postulated to be the relevant mathematical object that describes a quantum theory (because it contains observables in its self-adjoint part, and states as special ...


12

1) Configuration space is, in a sense, the possible "positions" of the mechanical system. The states of motion, eg. velocities/momenta are not part of the configuration space. The configuration space (especially when constraits are in the picture) is modelled as some real, $n$-dimensional differential manifold, which I'll denote as $\mathcal{C}$. The ...


12

Liouville's theorem says the accessible volume in phase space does not increase, but it tends to become narrow filaments that "fill up" a much larger volume. If you think of a particle in a reflecting box, you might start it with a known position $\pm 1$ mm in all three axes and a known velocity $\pm 1$ mm/sec in all three axes. This is a phase space ...


12

"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 integrals). This is in contrast to, e.g., path integrals (whose gauge fixing, Faddeev-Popov, and Fujikawa anomaly applications to gauge theories are virtually ...


12

The full groupoid of all (possibly non-linear) canonical transformations (CTs) is infinite-dimensional. Infinitesimal CTs are Hamiltonian vector fields $$ \delta z^I ~=~{\{F(z,t),z^I\}}, \qquad I~\in~\{1,\ldots,2n\}.\tag{1}$$ The symplectic group $Sp(2n, \mathbb{R})$ of dimension $n(2n+1)$ is the group of all linear time-independent CTs. The corresponding ...


11

I) Let us for simplicity work in 1D with $\hbar=1$. (The generalization to higher dimensions is straightforward.) Moreover, let us for simplicity take an operator $\hat{f}(\hat{X},\hat{P})$ without any ordering ambiguities, i.e., each monomial term in the symbol $f(x,p)$ depends only on either $x$ or $p$, but not on both. Then one possible motivation of ...


11

The Poisson bracket is the bracket of a Lie algebra defined by the symplectic 2-form. That's a lot to unpack so let's go through it slowly. A 2-form $\omega$ is an anti-symmetric two-tensor $\omega_{\mu\nu}$. If $\omega_{\mu\nu}x^\nu \neq 0$ at points where $x^\nu \neq 0$, then $\omega$ is said to be non-degenerate. So $\omega$ is like the metric tensor, ...


11

The most meaningful language to compare classical mechanics to quantum mechanics is the phase-space formulation of the latter, based on an invertible map between Hilbert space and phase-space: you want to compare apples with apples. In this formulation, a particle is represented not by a mere point in phase space, but, instead, by a constrained distribution ...


10

The name seems appropriate if consider that it probably comes from the case when the manifold is the cotangent bundle of a manifold. Then a point on $T^*M$ is a pair $(x,\alpha)$, where $x$ is a point on $M$ and $\alpha$ a one form. The definition of the tautological one form is: the value of the form at a point $(x,\alpha)$ on a tangent vector is obtained ...


10

The essential idea of a Poincaré map is to boil down the way you represent a dynamical system. For this, the system has to have certain properties, namely to return to some region in its state space from time to time. This is fulfilled if the dynamics is periodic, but it also works with chaotic dynamics. To give a simple example, instead of analysing the ...


10

i will try this one. A Hamiltonian system is (fully) integrable, which means there are $n$ ($n=$ number of dimensions) independent integrals of motion (note that completely integrable hamiltonian systems are very rare, almost all hamiltonian systems are not completely integrable). What this states in essence (and intuitively) is that the hamiltonian system ...


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