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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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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Constraints are not functional relations!

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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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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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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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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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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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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1answer
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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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1answer
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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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Quantization of an $\mathcal{c}$-algebra

I don't know if this is a valid question to ask, but I am wondering about the following: We are given a set of $\mathcal{c}$-number Lie-Brackets $$ [q_i,q_j] = 0= [p_i,p_j] \\ [q_i,p_j] = c_{ij}, $$ ...
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Addition of a constant to the operator due to quantization

Groenewold in his book On the Principles of Elementary Quantum Mechanics (1946, Springer Netherlands) page 45, maps the canonical momentum $p^2$ in the classical phase space to a general canonical ...
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Quantization of $c$-number Dirac-Bracket

I have a question concerning the quantization of phase-space variables $(q_1, q_2, q_3, p_1, p_2, p_3)$ with the Hamiltonian $$ H = \frac{3}{2}(p_1^2+p_2^2 +p_3^2) $$ and the following non-commuting ...
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Is The Seiberg-Witten Map Unique?

From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions $[x, y]=i \...
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Analogous structure of Diffusion and Schrödinger equation and definition of flux?

I came across some analogous structure of diffusion and the quantum mechanical particle (Schrödinger eq.). I have seen that there have been similar questions asked, but the (probablitily flux and the ...
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Validity of Canonical Quantization

I was studying about what does it mean canonical quantization treatment. But now I have the next question. Why if we establish canonical the commutation relations $$\left[q,p\right]=i\hbar,\quad \left[...
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How are the two definitions of Canonical Transformations related/equivalent? [duplicate]

I am aware of two definitions of canonical transformations which I state below. Definition $1$ We go from old set of $\{q_i,p_i,t\}$ of $2n$ phase space variables to a new set $\{Q(q_i,p_i,t),P(q_i,...
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Poisson bracket properties for tensor densities

I am doing some constraint analysis in an extended theory of gravity, and I am confused about Poisson brackets. The standard PB relations are for example $\{ab,c\} = a\{b,c\} + \{a,c\}b$ etc. But I am ...
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Change in Hamiltonian under infinitesimal canonical transformation [duplicate]

Consider an infinitesimal canonical transformation from the (symplectic) coordinates $z$ with Hamiltonian $H(z, t)$, to the coordinates $$ Z = z + \epsilon J \frac{\partial G}{\partial z} $$ with the ...
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Jacobian rules with canonical transformations

If we consider a canonical transformation from $(q,p)$ to $(Q,P)$, it is stated in several sources that by Jacobian rules, $$ \frac{\partial(Q,P)}{\partial(q,p)} = \frac{\partial(Q,P)/\partial(q,P)}{...
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Non-hamiltonian systems which evolve into hamiltonian by change of coordinates

I am very new to the subject, so please forgive my very naïf question. I learned that there are some non-hamiltonian systems which can become hamiltonian, just by a change of coordinates. I was given ...
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Geometric Intuition for the Moyal Product

I've recently been reading into deformation quantization as another formulation of quantum mechanics. I have focused on understanding the Moyal product in particular, as it contains the seeds for the ...
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1answer
133 views

Why/Do Hamilton's equations Hold with Complex Variables?

I am investigating the problem of taking a hamiltonian of bilinear terms, and converting them into a bunch of uncoupled oscillators, such as in a periodic lattice. To do this, you have to introduce ...
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Definition of weakly vanishing

In literature concerning dirac constraints, often the word weakly vanishing is used say for example, a function $f$ is called first class if its poisson brackets with all the constraints $\phi_i$ ...
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Confusion regarding properties of Poisson Brackets

I have just started learning about Poisson Brackets, and came across the following property $$\{q_i,q_j\}=0$$ And $$\{p_i,p_j\}=0.$$ Where $p$ and $q$ are respectively the momentum and position ...
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Poisson bracket of Hamiltonian with Hamiltonian always vanishes

Since Poisson bracket of Hamiltonian with Hamiltonian always vanishes then in case of explicit time dependence of Hamiltonian, how does Poisson bracket gives correct result?
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Poisson brackets of three dimensional angular momentum and its Lie algebra

I've recently noticed that the Poisson brackets of the three dimensional angular momentum $$\{L_i,L_j\}$$ in classical mechanics follow the same commutator relations as the standard basis of the Lie ...

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