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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Jacobi identity for Poisson bivector in local coordinates [closed]

This question was cross-posted from MSE, since I am not sure where it belongs better. Suppose a bivector field $\pi^{ij}$ such that $\pi^{ij}=-\pi^{ji}$, $\pi^{ij}\partial_{i}f\partial_{k}g=\{f, g\}$ ...
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Canonical commutation relation on the spatial boundary of the hypersurface

Consider the equal time commutation relation of a field given on a $d$ dimensional spacelike hypersurface $\Sigma$ of a $d+1$ dimensional manifold given by $$[\Pi(t, x), \Phi(t, x')] = i\hbar\delta^{(...
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Prove that $H'$ and $\Phi_a$, respectively defined by (8.207) and (8.208), are first-class functions (Nivaldo Lemos, problem 8.9.2) [closed]

I'm having trouble to solve this problem. In the book Analytics Mechanics, Nivaldo Lemos define two equations: Equation (8.207): $$ H' = H + \sum_{m=1}^M U_m\phi_m $$ Equation (8.208): $$ \Phi_a = ...
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The question about commutator $[\hat{x},\hat{p}]=i\hbar$ at $\hbar\rightarrow 0$ seemingly can't match with Poisson bracket $\{x,\,p\}=1$ [duplicate]

At the limit $\hbar\rightarrow 0$, all "quantum" should tend to "classical", but why is the quantum commutator $[\hat{x},\hat{p}]=i\hbar$ at $\hbar\rightarrow 0$ equal to $0$, but ...
a Fish in Dirac Sea's user avatar
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Poisson Bracket and commutators in quantum mechancs [duplicate]

how did they reach the conclusion that quantization of the Poisson brackets $ (A,B) $ was equal to the commutator $ \frac{1}{i\hbar}[A,B] $ in quantum mechanics? so the quantum equations of motion ...
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An example of constrained system [closed]

I have this homework on an analysis of a constrained Hamiltonian system and I need some help with the following problem. The system is described using Darboux coordinates and a specific Hamiltonian: ...
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Phase space of the $n$-vector model

The classical Heisenberg model is described in terms of the three-component unit vector $S_a(x)$, which is a function of position, $$H=\int d^dx\frac{1}{2}\sum_{a,i}\left(\partial_i S_a(x)\right)^2.$$ ...
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What goes wrong when we quantise a classical system without using $[X,P]=i\hbar$?

Let's say we have a classical system with a Poisson bracket. We quantise this system to get a quantum theory where we choose some variable to operator replacement : $x\rightarrow X, p\rightarrow P$, ...
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Commutation of the Hamiltonian with the generator of boost

Consider the Hamiltonian $H = (\textbf{P}^2+m^2)^{1/2}$ the generators of rotation and and boost given by $$M^{0i} = tP^i-x^iH \\ M^{ij} = x^iP^j-x^jP^i$$ where $x^i$ and $P^j$ satisfy $\{x^i, P^j\} = ...
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Derivation of self-dual gravity formulae

I am trying to read and understand this paper by Monteiro, Stark-Muchao, and Wikeley about self-dual yang-mills and self-dual gravity. In the introduction to this paper, they review a way to ...
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Canonical transformations in the covariant phase space formalism

As the title says, I'm looking for an explanation on how to apply canonical transformations when using the covariant phase space formalism. I'm familiar with the topic, but I haven't found a good ...
P. C. Spaniel's user avatar
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How to interpret Poisson bracket of fields in terms of causality?

In quantum field theory, the fact that space-like separated observables commute, i.e. $[\hat {\phi (x)}, \hat{\phi(y)}]=0$, is taken as the test for causality. The equivalent statement for classical ...
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Why is $[ \hat{A},\hat{B} ] \rightarrow i \hbar \text{{A, B}}$?

If we have two classical quantities $A$, $B$, and their corresponding quantum operators $\hat{A}$, $\hat{B}$, then their commutators and Poisson brackets are linked by $$ [ \hat{A},\hat{B} ] \...
Nicolas Schmid's user avatar
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$C^{\ast}$-algebra approach to classical mechanics

Can someone please help me understand how classical mechanics (for example in terms of Hamiltonian formalism) can be described in terms of $C^{\ast}$-algebras? I read usually that in this case the ...
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Unique distribution of charge on a conductor

If i place some charge on a conductor then it will distribute itself in such a way that electric field everywhere inside is zero. My text book says that only one kind of such charge distribution is ...
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How does Poisson bracket vanish in this equation?

I am studying the book "Lectures on Quantum Field Theory", by Ashok Das. I am stuck in the last step of equation (10.35) as I will explain below. Under section 10.2 (Dirac method and Dirac ...
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Calculating Poisson brackets in classical non-relativistic Hamiltonian field theory

Summary of the question: How can I prove the equal-time Poisson bracket relations for the classical Hamiltonian field theory? I.e $$[q(x,t),H(t)]_\mathrm{PB}=\dot{q}(x,t)\tag{1}$$ for a field $q$ and ...
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Questions on Symplectic approach to canonical transformations

Reading section 9.4 of Classical Mechanics by Goldstein, I got a question in my mind. That is, it says that for restricted canonical transformation, we have the new Hamiltonian equal to the old one. I ...
Ting-Kai Hsu's user avatar
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Non-symplectic Hamiltonian systems

I'm wondering when the phase space of a Hamiltonian system looses its symplectic structure. I think it happens when the Hamiltonian $H$ depends on a set of other variables $S_1,...,S_k$ as well as on ...
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Invariance of a volume element in phase space: What does it means?

I have been reading the third edition of Classical Mechanics by Goldstein, in particular, chapter 9 Poisson Brackets and Other canonical invariants. And it is shown that the magnitude of a volume ...
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Proof that canonical transformations implies symplectic condition [duplicate]

Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates $q,p$, Hamiltonian $H$, and new coordinates $Q(q,p),P(q,p)$, there exists a transformed Hamiltonian $K$ such ...
SecondOrderConfusion's user avatar
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What is the physical meaning of the minus sign in the Hamilton equation of momenta?

The Hamilton equations are $$ \dot{q}_k=\frac{\partial H}{\partial p_k}~~~~-\dot{p}_k=\frac{\partial H}{\partial q_k}~~~~~-\frac{\partial L}{\partial t}=\frac{\partial H}{\partial t}.$$ Does the minus ...
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How Poisson developed Poisson bracket?

In textbooks, they just give Poisson bracket used for restating the Hamiltonian in a powerful and elegant way. But why it is useful, and how Poisson created it?
professor T's user avatar
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Is my understanding of canonical transformations flawed?

Consider a system described by Hamilton's equations $$\dot{q}_i=\frac{\partial H}{\partial p_i}=\{q_i,H\}, \quad \dot{p}_i=-\frac{\partial H}{\partial q_i}=\{p_i,H\}.\tag{1}$$ I want to prove that a ...
Solidification's user avatar
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When do Hamilton's equation and Heisenberg equation give different solutions?

I'm self-studying QM and become fascinated about Heisenberg picture. I have a question about the relationship between Heisenberg picture and classical mechanics. Consider a simple form of Halmitonian: ...
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How does energy momentum tensor generate transformations?

I am reading https://arxiv.org/abs/hep-th/0008096 on p.20. Since I am not very familiar with the general canonical formalism, I have some trouble interpreting the following statements: Consider the ...
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The Poisson bracket for a Dirac field on an Eddington-Finkelstein background metric

I am interested in deriving the anti-commutation relations of a Dirac field for a $(1+1)$D Eddington-Finkelstein metric given by $$ ds^2 = f(r) dt^2 - 2 dt dr, \quad f(r) = 1 - \frac{r_s}{r}.$$ where $...
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Does the Hamiltonian formulation of classical mechanics require an inner product on physical space?

The Hamiltonian formulation of classical mechanics is quite broad and flexible; one of the only nontrivial physical assumptions that need to be made is that the degrees of freedom are continuous ...
tparker's user avatar
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Is there some notion of classical uncertainty which quantizes to quantum uncertainty?

I would like to know if there is some notion of classical uncertainty which quantizes to give quantum uncertainty? For instance, suppose we have a classical system whose phase space is given by a ...
leob's user avatar
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$ℏ$ in the canonical commutation relation

I am wondering what the physical meaning of the introduction of a "new" constant $\hbar$ in the CCR $[\hat{x},\hat{p}]=i\hbar$ is if you compare it to the classical Poisson-bracket $\{x,p\}=...
Python_Coder's user avatar
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Poisson brackets for a field theory

I'm performing a calculation involving Dirac constraints theory, and I need to calculate the Poisson brackets between constraints and the total Hamiltonian. The starting theory is described by a ...
Explosiveness's user avatar
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Commutator Constant

I have seen a lot of commutators in quantum mechanics having a constant factor $i\hbar$. I have read about Dirac supplanting Poisson Brackets with commutators having a constant $i\hbar$. I want to ...
Principia Mathematica's user avatar
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Canonical Transformation of Poisson Bracket [closed]

In Goldstein section 9.4(pg 381) it tells us that for a Hamiltonian that is not explicitly time dependent, transformations of $Q = Q(q,p), P = P(q,p)$ are canonical if $$\frac{\partial Q}{\partial q} =...
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Proof of Liouville's Theorem

The Wikipedia article on Canonical Transformations has a section on Liouville's Theorem. It makes the following argument: $$J=\frac{\partial(\mathbf{Q},\mathbf{P})}{\partial(\mathbf{q},\mathbf{p})}$$ ...
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Canonical Transformation "Indirect Approach"

The Wikipedia article on Canonical Transformations, in the section "Indirect Approach," makes the following argument: $$ \dot{Q}_{m} = \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \dot{\...
Jbag1212's user avatar
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Equivalence of symplectic condition and canonical transformation

In Goldstein's "Classical Mechanics", at page 384 it is claimed that given a point-transformation of phase space $$\underline{\zeta} = \underline{\zeta}(\underline{\eta}, t),\tag{9.59}$$ ...
Matteo Menghini's user avatar
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Is the expectation value of an operator obtained by canonical quantisation always the classical value? [closed]

I understand that generally, for some Hermitian operator $\hat{A}$, the classically measured value of a system is given by \begin{align} \langle \hat{A}\rangle=\langle\psi| \hat{A}|\psi\rangle \end{...
Adrien Amour's user avatar
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Partial derivative of momentum with respect to position in Poisson bracket representation

The representation of a Poisson bracket is given by the following equation: $$\tag{1} \{f,g\} = \sum_{s=1}^n \sum_{i=1}^{d=3}\left ( \frac{\partial f}{\partial x_i^{(s)}} \frac{\partial g}{\partial ...
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A problem understanding primary constraints meaning

I have some problems understanding the meaning of a function that vanishes weakly. As far as I can understand, when somebody writes that a function $F$ in the phase space vanishes weakly, that means ...
Thomas Belichick's user avatar
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Is there any classical correspondence of the Jacobi identity? [closed]

In QM, the commutator were closely related to the poison bracket, so much so that to promote the classical operators to the quantum operators were often associated as \begin{equation} \{A,B\} \text{(...
ShoutOutAndCalculate's user avatar
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Why Equal-time commutation relation?

Let $\phi(\mathbf{x},t)$ be a field, and $\pi(\mathbf{x},t)$ be the conjugate momentum field. A standard practice is to apply the equal time commutation relation: $$[\phi(\mathbf{x_1},t), \pi(\mathbf{...
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Why is the Lax-Jacobi identity useful in integrability?

I'm studying integrabilty and there is the so-called Lax-Jacobi identity, which is an implication of the classical Jacobi identity of the Poisson brackets of the Lax-Poisson structure: $$Cycl_{123} [...
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How to show that the Hamiltonian $H$ is invariant under flow generated by $F$?

I know usually if I have a transformation of phase space $Q(p,q), P(p,q)$ it is defined to be canonical if and only if its Jacobi matrix is part of the Symplectic Group or equivalently $\{Q^{i}, P_j \}...
Henry T.'s user avatar
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Classifying canonical transformation and scaling transformation

Lets assume we have a very simple transformation in 1 Dimension from $(x, p_x)\rightarrow (y,p_y)$ given as $$\begin{aligned} y &= cx \\ p_y &= c^{-1} p_x \end{aligned}$$ Is this a strictly ...
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Why is it useful to learn about Hamiltonian Mechanics in the framework of Symplectic Geometry?

...Other than providing a deeper insight into the mathematical background of dynamical systems. Does casting certain classes of problems in terms of symplectic geometry make solving them easier/...
Jason M Gray's user avatar
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3 answers
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Partial derivatives of canonical momenta in Poisson brackets

I will simply give an example for a general doubt about the Hamiltonian formulation. So, consider the spherical pendulum of length $l$ as an example of my perhaps more general question. The Lagrangian ...
Marsl's user avatar
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Why do we construct Lagrangian submanifolds after symplectic reductions

I am learning about Hamilton-Jacobi actions, symplectic reductions and Lagrangian submanifolds and I am trying to understand the relation between these concepts. I have read that Lagrangian ...
Spida's user avatar
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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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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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