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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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 $...
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Does the Hamiltonian formulation of classical mechanics require an inner product on physical space?
The Hamiltonian formulation of classical mechanics is quite broad and flexible; one of the only nontrivial physical assumptions that need to be made is that the degrees of freedom are continuous ...
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Is there some notion of classical uncertainty which quantizes to quantum uncertainty?
I would like to know if there is some notion of classical uncertainty which quantizes to give quantum uncertainty?
For instance, suppose we have a classical system whose phase space is given by a ...
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$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\}=...
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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 ...
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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 ...
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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{\...
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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 $$\underline{\zeta} = \underline{\zeta}(\underline{\eta}, t),\tag{9.59}$$ ...
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Coordinate transformation of an expression
I have an expression for $[F,G]_{\mathbf{p},\omega,\mathbf{R},T}$ as
$$
[F,G]_{\mathbf{p},\omega,\mathbf{R},T} =
\frac{\partial F}{\partial \omega} \frac{\partial G}{\partial T}
-\frac{\partial F}{\...
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Is the expectation value of an operator obtained by canonical quantisation always the classical value? [closed]
I understand that generally, for some Hermitian operator $\hat{A}$, the classically measured value of a system is given by
\begin{align}
\langle \hat{A}\rangle=\langle\psi| \hat{A}|\psi\rangle
\end{...
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Partial derivative of momentum with respect to position in Poisson bracket representation
The representation of a Poisson bracket is given by the following equation:
$$\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 ...
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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{(...
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Why Equal-time commutation relation?
Let $\phi(\bf{x},t)$ be a field, and $\pi(\bf{x},t)$ be the conjugate momentum field.
A standard practice is to apply the equal time commutation relation:
$$[\phi(\bf{x_1},t), \pi(\bf{x_2},t)] = i\...
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Why is the Lax-Jacobi identity useful in integrability?
I'm studying integrabilty and there is the so-called Lax-Jacobi identity, which is an implication of the classical Jacobi identity of the Poisson brackets of the Lax-Poisson structure:
$$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 \}...
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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/...
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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 ...
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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 ...
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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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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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A Question about Annihilation and Creation Operators in the 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:
$$\{L^i,L^j\}=\frac{\partial\epsilon^{\:i\:m}_{\:\:l}q^lp_m}{\partial q^k} \frac{\partial\epsilon_{\:\:s}^{\:j\:r}q^sp_r}{\...
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Doubts about General Relativity in Ashtekar variables and Thiemann trick - Perez "INTRODUCTION TO LOOP QUANTUM GRAVITY AND SPIN FOAMS"
I am working on the Perez book, INTRODUCTION TO LOOP QUANTUM GRAVITY AND SPIN FOAMS (https://arxiv.org/abs/gr-qc/0409061). In particular, I am interested in the understanding of the Poisson brackets (...
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Commutators as contour integrals in 2D CFT, and classical limits
In a 2D CFT, the commutator of two operators
$$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 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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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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