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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On-shell Poisson brackets and time derivative

In classical statistical mechanics, the information about a given system is given by a distribution of probability over phase space $\rho(p,q,t)$. Let $H(p,q, t)$ be the hamiltonian of the system and $...
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Prove that the Laplace Runge Lenz Vector is conserved using Poisson brackets [duplicate]

How do I show that the Laplace Runge Lenz vector $\mathbf A$ is conserved using Poisson brackets, where $$\mathbf A = \mathbf p \times \mathbf l + m\gamma \frac{\mathbf r}{|\mathbf r|}$$
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The different generators of canonical transformations

Consider the phase space of a one degree of freedom mechanical system. We can pass from one phase space coordinates to another phase space coordinates via a canonical transformation. I want to focus ...
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Canonical Realization of Poincare Symmetry of Dirac Spinor

I have a maybe stupid question about Noether charges and the Poisson bracket. If a classical field theory has a Poincare symmetry, then by using the Noether's theorem, one can write down its ...
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Integration by Parts in Liouville's Theorem

I am looking at a proof of Liouville's Theorem, which states that for $F, G \in C_0^\infty$ and a Hamiltonian $H$, the operator $$D_H = \sum_{i=1}^n\Big(\frac{\partial H}{\partial p_i} \frac{\partial}{...
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A derivation of the canonical commutation relations (CCR) written by Dirac?

Dirac in his book fundamental of quantum mechanic used the following derivation: Is this a derivation of the canonical commutation relations (CCR) in quantum mechanics?
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About characteristics of smearing function

I met the word smearing function (or test function) when I was learning ADM formalism in GR books. What makes me scratch my head is the reason of introducing such a smearing function when we calculate ...
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What is the correct general form of Hamilton's equation?

Usually, Hamilton's equations of motion are given by: $$ (1)\;\; \frac{dp}{dt} = -\frac{\partial H}{\partial q} \;\;\; \text{ and } \;\;\;(2)\;\; \frac{dq}{dt} =\frac{\partial H}{\partial p}.$$ ...
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What is the significance of commutator relationships in physics, e.g. $qp-pq = i \hbar$, $R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z$, etc?

Quantum mechanics has the commutator relationship: $$qp-pq = i \hbar$$ In relativity the Riemann tensor is a measure of how much covariant derivatives along a path commute. $$ R(X, Y)Z = \nabla_X\...
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What are the conserved currents and charges in QFT - operator form of Noether currents?

I can't figure out how to define/compute conserved charges and currents in Quantum Field Theory. I am following Peskin & Schroeder's Introduction to Quantum Field Theory, and in the second chapter ...
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What does it mean for two variables to be canonically conjugate?

The word "canonical" has been used in many of my classes (canonical ensemble, canonical transformations, canonical conjugate variables) and I am not really sure what it means physically. ...
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Functional derivative acts on covariant derivative

I'm confusing about how functional derivatives act on a covariant derivative. I'm doing such a calculation: In ADM formalism, let $h_{ij}(x)$ be the spatial metric while $\pi^{ij}(x)$ is its momentum ...
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Annihilation and creation operators in quantum harmonic oscillator

I'm new to quantum mechanics and I just have a doubt. If $\hat a$ is the annihilation operator of quantum harmonic oscillator and $ \hat a^{\dagger}$ is the creation operator, what is the value of $\...
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Problems with index position of angular momentum

I want to calculate the poisson bracket of two angular momentum components: $$\{L^i,L^j\}=\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial q^k} \frac{\partial\epsilon_{\:\:s}^{\:j\:r}q^sp_r}{\...
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Cross Product of two Hermitian Operators

The operator for linear momentum $\mathbf{p}$ and the operator for orbital angular momentum $\mathbf{L}$ ($\mathbf{L} = \mathbf{r} \times \mathbf{p}$) are Hermitian. Is the cross product between $\...
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Doubts about General Relativity in Ashtekar variables and Thiemann trick - Perez "INTRODUCTION TO LOOP QUANTUM GRAVITY AND SPIN FOAMS"

I am working on the Perez book, INTRODUCTION TO LOOP QUANTUM GRAVITY AND SPIN FOAMS (https://arxiv.org/abs/gr-qc/0409061). In particular, I am interested in the understanding of the Poisson brackets (...
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Commutators as contour integrals in 2D CFT, and classical limits

In a 2D CFT, the commutator of two operators $$A_i=\oint a_i(z)dz$$ can be given by $$[A_1,A_2]=\oint_0dw\oint_wdza_1(z)a_2(w)$$ where the $z$ integral is taken over a contour around $w$ and the $w$ ...
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Why is the Poisson bracket of the YM Hamiltonian with the secondary constraint zero?

Suppose we have the following Hamiltonian density $$\mathcal{H} = \frac{1}{2}\pi_i^a \pi_i^a + \frac{1}{4}F_{ij}^a F_{ij}^a $$ where $$F_{ij}^a = \partial_i A_j^a - \partial_j A_i^a + gf^{abc}A_i^b ...
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Poisson-bracket of linear/angular Momentum $\{p,L\}\,r$

I couldn't find it yet: simplifying $\{p,L\}\,r$ written out, plus assuming $\displaystyle r = q, \quad L = q\times p = \sum_{l =1}^{3}\varepsilon_{lmm} \hat{e}_l\,q_m\,p_n $ the breaket becomes: $\...
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Poisson bracket to quantum commutator for Grassmann-valued coordinates

In Dirac's Principles of Quantum Mechanics (pg. 86 eq. 5), the quantum commutator is motivated by looking for a bracket that satisfies the same properties of the Poisson bracket. When deriving the ...
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Liouville's theorem and uniform probability density

In Kardar's book on statistical physics it is claimed that Liouville's theorem gives support for the common assumption that the points in phase space compatible with the hamiltonian are all equally ...
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Isn't it just a co-incidence that the Poissson bracket and the commutator look similar? [duplicate]

Both formulas describe the rate of change of some observable and both look kinda similar. However, the Poisson bracket formula is arrived at by the chain rule of derivatives (by differentiating $w(q,p)...
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How to derive infinitesimal gauge transformations from constraints?

I am reading some papers about quantizing the gravitational fields, for example, here, here, and here. Since the classical actions for gravitational fields are singular, they contain some constraints. ...
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Generating Functions for Extended Canonical Transformations

From Goldstein we have that, for non-extended ($\lambda =1 $), the generating function of third type is $$F = F_3(p,Q,t) + q_ip_i.\tag{1}$$ Although I found it hard to see if that would hold true also ...
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Is it really necessary for a Hamiltonian density to vary with a total derivative under symmetries?

I am studying the effect of infinitesimal transformations on a hamiltonian field theory, $\mathscr{H}(\mathscr{Q}_n,\mathscr{P}_n,\partial_i\mathscr{Q}_n,x)$, with scalar fields and momenta $\mathscr{...
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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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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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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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1 vote
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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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2 votes
2 answers
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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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2 votes
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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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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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2 votes
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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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