Questions tagged [canonical-conjugation]
62 questions
33
votes
9 answers
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.?
user6090
- 1,446
19
votes
2 answers
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
derpy
- 341
16
votes
4 answers
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 fact that two variables are a conjugate pair to…
jheindel
- 1,059
13
votes
4 answers
Relation between energy and time
I would like help in understanding something that has been causing me a lot of trouble recently: Why is energy always related to time in physics?
Examples include the 4-momentum, the energy-time uncertainty principle and Hamiltonian mechanics.…
Xirven
- 437
12
votes
4 answers
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.
More specifically, in the context of the Hamiltonian…
realanswers
- 523
9
votes
1 answer
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 mirrors; he gave a good explanation during the…
Emilio Pisanty
- 137,480
8
votes
1 answer
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?
Fahad Abbasi
- 97
7
votes
2 answers
What does "conjugate variable" encompass, and how does it do this?
Context for the question: I am studying Legendre transforms in statistical mechanics. I have read that the Legendre transform allows us to transform between conjugate variables in expressions. For example, if I begin with $E(S,V,N)$, a Legendre…
Relativisticcucumber
- 1,987
6
votes
1 answer
Canonical quantisation of conjugate Dirac field
This may be a stupid question to ask. For Dirac field, we know the Lagrangian $${\cal L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi \tag{1}$$ is not symmetric in $\psi$ and its conjugate field $$\bar{\psi}=\psi^\dagger\gamma^0,\tag{2}$$ so that…
SkyFucker
- 143
- 3
6
votes
1 answer
Non-hermiticity of Dirac Lagrangian: null momentum?
The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are
$$
\pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\
\bar \pi=\frac{\partial L}{\partial \bar\psi_{,0}}=0
$$
The fact that $\bar\pi=0$…
AccidentalFourierTransform
- 56,647
5
votes
0 answers
Lagrangian with vanishing conjugate momentum, independent variables
Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes:
$$\pi_n=\frac{\partial\mathcal L}{\partial(\partial_0\phi_n)}=0.$$
What…
LYg
- 1,201
5
votes
3 answers
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 rather than discrete.
In the Hamiltonian formalism…
tparker
- 51,104
5
votes
2 answers
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 if this is possible, how does one construct…
e4alex
- 73
- 4
5
votes
1 answer
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 fundamental thermodynamic equations, all the…
Daddy Kropotkin
- 2,790
5
votes
2 answers
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 think of are the quantities of thermodynamics -…
Lo Scrondo
- 647