Questions tagged [canonical-conjugation]

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Thermodynamical conjugate variables

In thermodynamics the potentials are typically only a function of 2 variables, say $$U=U(S,V)$$ with entropy $S$ and volume $V$. I see that conjugate pairs $S,T$ or $p,V$ always have the unit of ...
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42 views

Transformation to canonically conjugate coordinates in Special Relativity

Just like we can Fourier transform a field from $$x^\mu = (ct, x,y,z) \rightarrow p^\mu = (E/c, p_x, p_y, p_z)$$ via a Fourier transform, for spherical coordinates can we Fourier transform in the same ...
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0answers
32 views

Can the Fock-Schwinger (radial) gauge condition be written as momentum space divergence?

The Lorenz Gauge can be written (in QED) as $\partial^{\mu}A_{\mu} = 0$ or equivalently as $p^{\mu}A_{\mu} = 0$. The Fock-Schwinger gauge is similar: $x^{\mu}A_{\mu} = 0$. Can it be written ...
5
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1answer
220 views

When can one find a canonically conjugate operator?

Suppose one is given a self-adjoint operator $A$ acting on an infinite dimensional separable Hilbert space $\mathcal{H}$. Under what conditions can one find an operator $B$ such that $[A,B] = i$? And ...
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0answers
78 views

Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian

I want to show that the gauge transformation $$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$ corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
5
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1answer
134 views

Origin of conjugate variables in physical theories

Why do conjugate variables come in pairs? For example, in classical mechanics we have the generalized coordinates of position and momentum, and there is Jacobi's action-angle coordinates. Also, in the ...
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1answer
90 views

Four special types of canonical transformations

Let $(q,p) \mapsto (Q,P)$ be a diffeomorphism of phase space. Then this is a canonical transformation (CT) if $$p\dot{q}-H(q,p,t)=P\dot{Q}-K(Q,P,t) + \frac{dM}{dt}\tag{1}$$ for some $M=M(q,p,Q,P,t)$....
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3answers
437 views

Planck's constant and the Uncertainty principle

Why should the uncertainty in measurement of two conjugate variables, say $p$ and $x$, be of the order of Planck's constant or higher and not lower? What is so sacrosanct about $h$? I always think $h$ ...
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2answers
138 views

Is there no coordinate conjugate to $L_x$?

The conjugate momentum corresponding to $\phi$ (azimuthal angle in sp. polar coordinate) is $L_z$ (sometimes written $L_\phi$) which is frequently used in quantum mechanics. Why is there no coordinate ...
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76 views

Do we need to know first time derivatives of Klein-Gordon fields to make future predictions?

In the quantisation of the Klein Gordon field, because it has a second derivative, we need to know the values of the field and also the first derivative in order to predict future values of the field. ...
4
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2answers
651 views

Conjugate variables in thermodynamics vs. Hamiltonian mechanics

According to Wikipedia, the canonical coordinates $p, q$ of analytical mechanics form a conjugate variables' pair - not just a canonically conjugate one. However, the "conjugate variables" I directly ...
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0answers
203 views

Time-dependent Canonical transformation

Suppose that the Hamiltonian of the mechanical system under analysis depends on two complex conjugate variables $a$ and $a^*$, so we have: $$ H=H\left(a,a^*\right) $$ Hamilton equations read $$ ...
9
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1answer
159 views

In what sense are shot noise and photon pressure canonically-conjugate variables in the LIGO interferometer?

This week I saw a seminar by Kip Thorne and he mentioned that in the LIGO interferometer, the photon shot noise is actually canonically conjugate to the noise induced by photon pressure acting on the ...
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0answers
311 views

Why is entropy not a thermodynamic potential in the sense of Legendre transform?

We can formulate thermodynamic theory using entropy S as state function or equivalently using the internal energy U, and U is related with all other thermodynamic potentials via the Legendre transform,...
7
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2answers
836 views

How does one know if two variables are conjugate pairs?

First of all, I am having a hard time finding any good definition of what a conjugate pair actually is in terms of physical variables, and yet I have read a number of different things which use the ...
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0answers
110 views

Is it always possible to define/find conjugate variables? And if yes how one can find it?

My question is in the context of both classical and quantum mechancis and field theory. Generally, how can one define/find the (canonically) conjugate of some variable/operator/field? Examples ...
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2answers
185 views

Unitary translation in phase space coordinate

If we suppose that we can translate one point to another point in phase space $(x,p)$ through the following operators, $$T(\Delta x) = \exp(-i p~\Delta x ) $$ and $$T(\Delta p) = \exp(-i x~\Delta p ) ,...
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1answer
158 views

Key Assumption to derive the Uncertainty Principle — Canonical Conjugate Operators [closed]

From this related question, Rigorous Mathematical Proof of the Uncertainty Principle from First Principles The key assumption to derive the uncertainty principle seems to be the relationship ...
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1answer
108 views

Placement of indices in canonical commutation relations of coordinates and conjugate momenta as well as fields and conjugate momenta

The canonical commutation relations between generalised coordinates $q_a$ and their conjugate momenta $p^a$ are given by $[q_a,q_b]=[p^a,p^b]=0$ $[q_a,p^b]=i\delta^b_a$. Furthermore, the canonical ...
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2answers
630 views

What is the momentum canonically conjugate to spin in QM?

In Kopec and Usadel's Phys. Rev. Lett. 78.1988, a spin glass Hamiltonian is introduced in the form: $$ H = \frac{\Delta}{2}\sum_i \Pi^2_i - \sum_{i<j}J_{ij}\sigma_i \sigma_j, $$ where the ...
5
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1answer
253 views

Non-hermiticity of Dirac Lagrangian: null momentum?

The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are $$ \pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\ \bar \pi=\frac{\partial L}{\...
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1answer
155 views

What is the difference between the momentum in the Fourier transform of a scalar field and the conjugate momentum of the field?

What is the difference between the momentum $p$ in $e^{i\mathbf{p}\cdot{\mathbf{x}}}$ in the Fourier transform of a scalar field and the corresponding conjugate momenta $\pi(x)$ of the scalar field?
5
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2answers
935 views

Energy and momentum as partial derivatives of on-shell action in field theory

According to L&L, if we fix the initial position of a particle at a given time and consider the on-shell action as a function of the final coordinates and time, $S(q_1, \ldots, q_n, t)$, then... $...
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0answers
217 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: $$\pi_n=\frac{\partial\...
3
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1answer
925 views

Why is momentum (instead of something else) the canonical conjugate of position?

Why did nature decide to make conjugate of position to be momentum? Since energy and position do not commute, why not energy? What determines the pairing of time with energy and momentum with position?...
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8answers
5k views

Why do quantum physical properties come in pairs?

Why do quantum physical properties come in pairs, governed by the uncertainty principle (that is, position and momentum?) Why not in groups of three, four, etc.?