Questions tagged [poisson-brackets]

In phase-space classical Hamiltonian mechanics, the Poisson bracket is an antisymmetric binary operation acting like a derivative.

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Prove that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable

How do I show that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable, $\hat{H}(\hat{x},\hat{p})$ is the Hamiltonian of the system and $\hat{...
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Operator ordering (for the Groenewold-van Hove theorem)

Suppose we have the (un)usual Schrödinger representation $\pi'(\cdot)$ of the Heisenberg algebra, along with the extension in $\mathbb{sl}(2,\mathbb{R})$ for the quadratic polynomials. It is assumed ...
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Is the definition of a canonical transformation symmetric in the specification of old and new coordinates?

Consider the following transformation: $$q=P^\alpha \cos(\beta Q)$$ $$p=P^\alpha \sin(\beta Q)$$ for $\alpha=1/2$ and $\beta=2$. Now, by convention, one takes $(q,p)$ to be the old coordinates and $(Q,...
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System vector, angular momentum, and rotation

In Goldstein 3rd edision p. 409, it is stated that $$\partial \mathbf{F}=[\mathbf{F}, \mathbf{L}\cdot\mathbf{n}]=\mathbf{n}\times\mathbf{F}\tag{9.121}$$ if $\mathbf{F}$ is a function of system ...
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How can you confirm that two variables are canonically conjugate using Poisson brackets?

Suppose you have two conjugate variables $q$ and $p$ that are canonically transformed into two other variables $Q$ and $P$. What needs to hold true for these variables in terms of Poisson brackets? I ...
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verify canonically conjugate variables by way of Poisson brackets

How do I verify using Poisson brackets that a set of variables are canonically conjugate? It's for a homework problem but I'm interested in a general approach/reference if possible. We are using ...
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Qualitatively connecting classical Poisson brackets and quantum commutators

I have read many answers on the correspondence between classical Poisson brackets and quantum commutators, but I was wondering if it was able to be spelled out in simple words. In this answer (What is ...
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1answer
54 views

Jacobi identity of the anti-bracket

I'm currently reading a volume 2 of Weinberg's QFT, and am puzzled by the Jacobi identities of the anti-bracket.  The anti-bracket is defined using the anti-field $\chi^n$ and $\chi_n^{‡}$ as follows $...
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What's the importance of Poisson brackets? [duplicate]

$$F=F(q,p,t)$$ $$\frac{dF}{dt}=\frac{\partial F}{\partial q}\frac{\partial q}{\partial t}+\frac{\partial F}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial F}{\partial t}$$ $$=\frac{\partial F}...
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Poisson algebra and the origin of canonical quantization

Professor Achim Kempf in his lecture note mentioned that non-commutativity of quantum observables in the associated Poisson algebra to the system, impose CCR It was Dirac who first realized that all ...
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Poisson algebra between Hamiltonian constraints in the connection formulation of the loop $f(R)$ gravity

I have a problem with calculation of the Poisson algebra for Hamiltonian in loop $f(R)$ gravity, which is given by: with the smeared version $\int _\Sigma d^3 x N H$. $\tilde{K^i_a}$ is the extrinsic ...
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Commutation relation in Euclidean time

Does the usual commutation relationship $$[q,p]=i\hbar\tag{56}$$ change to $$[q,p]=\hbar\tag{55}$$ when making a Wick rotation to Euclidean time? and if so, what is the physical reason to ...
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232 views

Poisson bracket of the angular momentum and a scalar function

In the context of the Hamiltonian mechanics, I am trying to demonstrate the following statement: For any scalar function $f$, just as the dot product $\boldsymbol{q}·\boldsymbol{p}$, the Poisson ...
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1answer
41 views

Odd or Even symplectic structure in BV formalism

I am studying Batalin-Vilkovisky formalism. I am a little bit confused on what an odd (or even) symplectic structure is (i know what the degree of the underlying 2-form is). I can not find a clear ...
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Poisson brackets of bosonic string oscillators

I'm reading string theory books and I'm stuck at the moment when we consider a Hamiltonian version of the classical string. Namely I don't understand how to derive the Poisson brackets for string ...
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Intuition behind Poisson bracket of Arbitrary Function, & dot product of Angular Momentum & Constant Vector being the Cross Product of the Latter Two

In my introductory mechanics class, we were given a home assignment to calculate the following Poisson bracket. In $\mathbb{R}^3$, let $\vec{r}$ and $\vec{p}$ be the generalized coordinate and the ...
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Strong equality in Quantization of Gauge Systems by Henneaux and Teitelboim

I am new to the concept of weak and strong equalities, and I have a doubt trying to derive an expression. In section $1.2.1$ of Henneaux and Teitelboim's Quantization of Gauge Systems, there is a ...
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Commutation relations inconsistent with constraints

In section $9.5$ of Weinberg's Lectures on Quantum Mechanics, he uses an example to explain the clasification of constraints. The Lagrangian for a non-relativistic particle that is constrained to ...
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1answer
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Constraints are not functional relations! [closed]

I am reading a Wikipedia article on Dirac brackets. At the bottom of the page "illustration on example provided" the article states that for a system with constraints: $$ \phi_1 = p_x + \...
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Why are first class constraints harder to quantize than second class constraints?

I understand that the well known system with the second class constraints: \begin{align} &q_1 = 0 \\ &p_1 = 0 \end{align} has the apparent problem when performing quantization using the ...
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Is it in general? $[\Lambda,\Omega]=i\hbar\{\lambda,\omega\}_{x,p}$ [duplicate]

In R. Shankar's book, He has written $$[X_i,P_j]=i\hbar\{x_i,p_j\}=i\hbar$$ Is there any specific reason to use the Poisson bracket? Is there any general relation which looks like? $$[\Lambda,\Omega]=...
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Canonical transformations - sufficient & neccessary argument

I see in many textbooks that for a transformation of coordinates $P=P(q,p,t), Q=Q(q,p,t)$ it is sufficient & neccessary to check: $$[Q_i,Q_j]_{q,p} = 0$$ $$[P_i,P_j]_{q,p} = 0 $$ $$[Q_i,P_j]_{q,p}...
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Why there is no commutator term in the pre-sympletic density?

In this post I'm considering the Covariant Phase Space (CPS) formalism as presented by Lee & Wald in "Local symmetries and constraints ". In the CPS formalism we take the Lagrangian form ...
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162 views

Are Poisson brackets preserved during a canonical transformation?

Fix a Hamiltonian $H(q, p, t)$. Definition: A transformation $(q, p, t)\mapsto (Q(q, p, t), P(q, p, t), t)$ is said to be canonical iff for the Kamiltonian $K$ defined as $H(q, p, t)=K(Q(q, p, t), P(q,...
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The relationship between symplectomorphism, canonical transformations, and the symplectic group

This is a follow up to this question. In the answer by Qmechanic, they state that the symplectic group, $Sp(2n,\mathbb{R})$, is the group of linear, time-independent canonical transformations. If we ...
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66 views

Two consecutive symmetry transformation generated via Poisson brackets

Question If an infinitesimal symmetry transformation parametrized by Killing field $f^\mu(x)$ $$ \delta_f\phi=\phi'(x)-\phi(x)=f^\mu\partial_\mu\phi\tag1 $$ can be generated via Poisson bracket $$ \...
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How to evaluate the equal time Poisson bracket $\{ \phi(x), \vec{\nabla}_y\phi(y) \cdot \vec{\nabla}_y\phi(y)\}$?

I learned that for a classical scalar field theory in 4 dimensions, we can use the equal time Poisson brackets $$\{ \phi(x), \phi(y) \}_{x_0=y_0}=0$$ $$\{ \pi(x),\pi(y)\}_{x_0=y_0} =0$$ $$\{\phi(x),\...
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Particle in the Yukawa potential - Showing that the $z$-component of the angular momentum is conserved

I'm sorry for this homework question but I'm sitting a really long time now on this rather "easy" looking problem and I can't find a way to solve it. I'm given the Hamiltonian of the ...
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Explicit independence of Hamiltonian phase-space variables from the time parameter

In general, we have for a Hamiltonian flow $H$ of some "time" parameter $t$, the following relation for any function $f=f(q,p;t)$ of the phase-space generalized position ($q$) and conjugate ...
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About the Lie algebra of the angular momentum Poisson bracket structure [duplicate]

The components of the classical angular momentum $L_i$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi ...
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How to evaluate Commutator Bracket $\left[x,\frac{\partial}{\partial x}\right]$ indirectly using Poisson Bracket?

I want to evaluate $\left[x,\frac{\partial}{\partial x}\right]$ using a Poisson bracket. Can this be done? I have heard that the commutator bracket is $i\hbar$ times the Poisson bracket. I tried to do ...
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28 views

Liouville CFT Poisson Brackets

I have been given an action of the form: $$S = \frac{1}{4\pi}\int d^2\sigma \ \sqrt{-g}\left(\frac{1}{2}\partial_\mu\phi \partial^\mu\phi + \frac{1}{\zeta}\phi R + \frac{\mu}{2\zeta^2}e^{\zeta\phi} \...
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204 views

Proof of Poincaré algebra with Poisson bracket

I'm trying to prove that the generators of Poincaré group in Poisson bracket close the well-known Poincaré algebra. In particular, I don't know how to prove that $\{P_\mu,P_\nu\}=0$. Let's take an ...
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134 views

Poisson Bracket in Relativistic Field Theory

In QFT when we wrote a Lagrangian for a classical field, we switched to Hamiltonian formulation and introduced Poisson Bracket as $$\{A(x,t),B(y,t)\}^{(3)} = \int_{\Sigma_t}d^3z \left( \frac{\delta A(...
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An infinitesmal transformation that is canonical

The following infinitesimal transformation of phase space coordinates (for infinitesimal $\epsilon$) is apparently canonical (preserving Hamilton's equations and Poisson brackets): $$ q_i' = q_i + \...
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Why do symplectic form need to be closed in classical mechanics? [duplicate]

Mathematical structure of classical mechanic is described by symplectic geometry which is a smooth manifold with a non-degenrate closed 2-form $\omega$. I understand the requirement that $\omega$ ...
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94 views

Example of time-dependent constant of motion in classical mechanics

In classical mechanics text, when learning about Poisson brackets, one gets $\frac{df}{dt} = \{f,H\} +\frac{\partial f}{\partial t}$, where $H$ is the Hamiltonian of the system and for $\frac{df}{dt}=...
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Poisson-Bracket representation of the Poincaré group and symmetries of dynamical systems

In canonical formalism we know that a symmetry for the dynamical system can be expressed by $\{H,f\}=0$, where $H$ is the hamiltonian of the system and $f$ is the smooth function associated to the ...
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Second quantization with equal time Poisson bracket

One of the canonical prescriptions to quantize a classical theory is to reproduce the *-Algebra of functions (functionals) on $M$ on a complex Hilbert Space. In relativistic quantum mechanics I saw ...
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Is it possible to minimize the number of axioms/rules of the canonical quantization?

In the standard canonical quantization procedure there are two rules. Transform all quantities to operators. Transform the Poisson bracket to a commutator. Of course it will be nicer to minimize the ...
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Algebra of Noether's charges and algebra of symmetry transformations

I'm trying to understand the connection of algebra of transformations under a commutator and algebra of Noether's charges under Poisson bracket. I have a problem that results I infer from theoretical ...
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How to use the Poisson Bracket to generate a finite Lorentz transformation?

I'm doing something very wrong or It seems to me that I can't generate a finite Lorentz transformation using the exponential of the infinitesimal Lorentz boosts. Let me define $L_{x;v}$ as the ...
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Derivation of Poisson bracket and commutator of position and conserve charge [closed]

How can I prove these two relations? Assuming $\mathbf{D}$ is defined as $$\mathbf{D}=\sum\frac{\partial\mathcal{L}}{\partial\dot{x}_i}\delta x_i-\mathcal{L}=\mathcal{H}t-\frac{1}{2}\mathbf{p}\cdot\...
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Understanding the Functional Poisson Bracket

In classical field theory (for a single field $\psi$) the dynamical variables are defined to be functions of the fields $\psi$, $\pi$, $\partial_{x_{i}}\psi$ and maybe $\mathbf{r}$, where $\pi$ is the ...
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Time evolution of Galilean boost

I was introduced the generator of Galilean boost $K=mx-pt$. I was given an Hamiltonian with several particles: $H=\sum_i \frac{p_i^2}{2m_i}+V(|x_i-x_j|)$ where the potential only depends on the ...
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How to obtain commutation relations from symplectic potential?

I am studying the notes on susy qm in 1 dimension of David Skinner (http://www.damtp.cam.ac.uk/user/dbs26/SUSY.html) (which itself follows the mirror symmetry book by Vafa and Hori (relevant pp. 206 - ...
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Poisson Bracket $\{\delta_{ij}, g\}$ and partial derivative of Kronecker delta

I am currently working through Shankar's Princeiple of Quantum Mechanics Exercise 2.8.2 is to verify that the infinitesimal transformation generated by any dynamical variable g is a canonical ...
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Poisson Bracket of $\{Q, P\}$ in the original coordinate $(q, p)$

For simplicity, I use $(q,p)$ and $(Q,P)$ instead of $(q_i,p_i)$ and $(Q_i,P_i)$. I know that we should get $\{Q, P\} = 1$ for a canonical transformation $(q,p)\rightarrow(Q,P)$. But we also know from ...
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What is the "secret " behind canonical quantization?

The way (and perhaps most students around the world) I was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
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Physics that calls for deeply nested Lie/Poisson brackets

I've been scouring physics for non-associative situations, particularly where study of quasigroups and loops might come in handy (they always seem to be left out). The poisson and lie brackets form a ...

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