# Tag Info

## Hot answers tagged poisson-brackets

### What is the "secret " behind canonical quantization?

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 ...
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### The quantum analogue of Liouville's theorem

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 ...
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### Hamilton equations from Poisson bracket's formulation

The confusion stems from a bit of mathematical sloppiness. It's necessary useful sloppiness, because as you'll see in a moment, the full machinery is a pain, but it's helpful to see and keep in the ...
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### Why Equal-time commutation relation?

The problem is that once you define your operators at a fixed time $t_0$, your operators at all later times are immediately determined by the Heisenberg equations of motion. So you are not free to ...
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### Is a canonical transformation equivalent to a transformation that preserves volume and orientation?

In dimension $2n>2$ they are not equivalent since (for time-independent transformations) canonical is equivalent to $$\sum_{k=1}^n dq^k\wedge dp_k = \sum_{k=1}^n dQ^k\wedge dP_k\tag{1}$$ whereas ...
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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 ...
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OP's proposal \begin{align} [\hat{\phi}({\bf x}_1,t_1),\hat{\phi}({\bf x}_2,t_2)]~=~&0,\cr [\hat{\phi}({\bf x}_1,t_1),\hat{\pi}({\bf x}_2,t_2)]~=~&i\hbar\hat{\bf 1}~\delta^3({\bf x}_1-{\bf ... • 207k 11 votes Accepted ### Non-Euclidean Phase Space? Given a symplectic manifold (M,\omega), it is natural to ponder what tangent bundle connection\nabla: \Gamma(TM)\times\Gamma(TM)\to \Gamma(TM) \tag{1}$$to chose? Generically, it is natural to ... • 207k 10 votes Accepted ### What is the physical meaning of the minus sign in the Hamilton equation of momenta? Consider the Hamiltonian for a single particle in some potential$$H(p,q) = \frac{p^2}{2m} + V(q)$$where the first term is the kinetic energy and the second term is the potential energy. In this case,... 9 votes ### Is a canonical transformation equivalent to a transformation that preserves volume and orientation? Counterexample: The transformation$$Q^1~=~2q^1 ,\qquad P_1~=~p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~p_2 $$preserves phase space volume & orientation, but is not a symplectomorphism.^1 ... • 207k 9 votes Accepted ### Why are first class constraints harder to quantize than second class constraints? First-class constraints generate gauge transformations (assuming the Dirac conjecture), i.e. map physically equivalent states onto each other. Even if you do not assume the Dirac conjecture, then ... • 126k 8 votes ### Motivation for covariant phase space Given a dynamical system \dot x=F(x) with continuously differentiable F, there is a canonical bijection between initial conditions at a fixed time and solution trajectories. If the dynamical ... • 45.3k 8 votes ### Proof of constructing action-angle coordinates on Hamiltonian system OP's actual question follows directly from Theorem 5.3 in Ref. 2, but that leaves the obvious question: How to prove Theorem 5.3? It seems the only really satisfying answer would be to outline a ... • 207k 8 votes ### Does the Heisenberg equation for fields and canonical momentums hold as well for the hamiltonian density operator instead of the Hamiltonian operator? The answer is No. For starters for dimensional reasons. A density carries dimension L^{-3}. In the classical (as opposed to the quantum) case, it is tempting to (at least partially) incorporate OP'... • 207k 8 votes ### Why do we care only about canonical transformations? See What's the point of Hamiltonian mechanics? for a closely related question. The answers in that thread mention various benefits with the Hamiltonian formulation, among other things: Analysis ... • 207k 8 votes ### What is the "secret " behind canonical quantization? Instead, we were given a recipe how to quantize a classical theory, which is based on the rule of transforming all quantities to operators, and that Poisson bracket is transformed to a commutator. For ... • 64.2k 8 votes ### What does it mean for two variables to be canonically conjugate? One can see canonical conjugate variables as a pair of variables which generate the displacement of each other. For instance, consider the quantum mechanical case$$ [\hat{x},\hat{p}]=i\hbar $$If we ... • 480 8 votes Accepted ### ℏ in the canonical commutation relation Note that while the commutator [\cdot,\cdot] is dimensionless, the Poisson bracket \{\cdot,\cdot\} carries dimension of inverse angular momentum. So a quantity of dimension of angular momentum is ... • 207k 7 votes ### Dequantizing Dirac's quantization rule The statement is true by the very definition of quantization, i.e. there is nothing to show. So let's talk about the definition of quantization, which is a map from classical observables to quantum ... • 126k 7 votes ### What is the physical meaning of the minus sign in the Hamilton equation of momenta? More broadly speaking, the asymmetric minus sign in Hamilton's equations is intimately related to the antisymmetry of the symplectic/Poisson structure, which in turn has several consequences, e.g. ... • 207k 6 votes Accepted ### Finding higher order first integrals of a Hamiltonian The simple answer, overall, is that you always have to assume an integral of motion in a certain form and look if the conditions for its existence are even fulfilled. Generically, they are not, and if ... • 20.4k 6 votes Accepted ### Whence the i in QM Poisson bracket definition? The imaginary unit i is there to turn quantum observables/selfadjoint operators into anti-selfadjoint operators, so that they form a Lie algebra wrt. the commutator. Or equivalently, consider the ... • 207k 6 votes ### Hamilton equations from Poisson bracket's formulation The "partial" time derivative \frac{\partial}{\partial t} means in this context an explicit time differentiation. A function$$(q,p,t)~\mapsto~ f(q,p,t)$$of phase space and time is said to have an ... • 207k 6 votes ### Why do we care only about canonical transformations? You're right. Sometimes it can be fine to take a non-canonical transformation and this is often done in practice (though not usually emphasized). I'll explain in an example from quantum mechanics. The ... • 14.4k 6 votes ### Poisson brackets of three dimensional angular momentum and its Lie algebra The jth component of the angular momentum L_j is the Noether charge for 3D rotation around the jth axis. It satisfies the so(3) Lie algebra$$\begin{align}\{L_j,L_k\}~=~& \epsilon_{jk\ell}...
A factor $i$ is missing in your first equation on the RHS (a minor point). The dimensions of the LHS and the RHS of your second equation differ, so this relation must be wrong already for this very ...