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Mar
1
comment Is there a Lagrangian formulation of statistical mechanics?
@Nathaniel when $x_0$ and $x_1$ are the boundary points of a classical trajectory (satisfying the Hamilton's equations), then $S$ is the value of the classical action. Of course we can use this function for any two points on the configuration space not necessary along a classical trajectory. In this case the Hamilton Jacobi-phase function generates canonical transformations (for this reason, some authors call it the generating function). The basic application that I know of this function is in the solution of the Hamilton's equations given boundary and not initial conditions.
Mar
1
comment Is there a Lagrangian formulation of statistical mechanics?
@Vibert Spin systems can also be formulated on phase space, for example a single spin phase space can be chosen to be a 2-sphere (the collection of all spin directions). This description can also be generalized to the case of many spins, which can be formulated on "non-flat" phase spaces. The phase space is just the collection of initial data of an evolving mechanical system; The Hamiltonian contains the information about the interaction. Thus, the same description applies for both the weakly and the strongly interacting cases.
Mar
1
awarded  Revival
Feb
28
answered Is there a Lagrangian formulation of statistical mechanics?
Feb
27
answered Form of the Classical EM Lagrangian
Feb
27
answered Topological Charge. What is it Physically?
Feb
24
awarded  Yearling
Feb
18
answered Spontaneous breaking of Lorentz invariance in gauge theories
Feb
1
awarded  research-level
Jan
26
awarded  Enlightened
Jan
26
awarded  Nice Answer
Dec
30
comment Path integral with zero energy modes
@Greg I can't see your E-mail, instead I placed the copy in a file exchange server, fileconvoy.com/… where it will be abailable in the next 7 days
Dec
28
comment Path integral with zero energy modes
@Greg, 1) The unitary transformation U does not need to preserve $\mathrm{ker}(D)$, sorry for not emphasizing that one must project on the top form after performing the action, because the top form subspace is one dimensional. I'll try to add in a few days an explicit computation of the scalar multiple and the cocycle condition. 2) I'll be happy to help if you need me to send a copy of the article.
Dec
26
answered Path integral with zero energy modes
Dec
23
comment Classical vs. Quantum use of the spin 4-vector
Sorry for the late response, I added an update explaining the ultra-relativistic case.
Dec
23
revised Classical vs. Quantum use of the spin 4-vector
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Dec
21
comment Classical vs. Quantum use of the spin 4-vector
The boost transformation of the spatial components of the Pauli Lubanski vector is the third equation copied from the reference article. For a massive particle, the spin is the value of the spatial component vector at its rest frame. Knowing the momentum and the Pauli-Lubanski vector, one can perform a boost to a frame where the particle is at rest and get its spin. In a frame where the particle is not at rest the spatial components of the Pauli Lubanski vector still satisfy the spin commutation relations since they are composed from the spin and an angular momentum, thus should be quantized.
Dec
20
comment Classical vs. Quantum use of the spin 4-vector
Basically, yes, but please notice that the spatial spin components in the rest frame satisfy the angular momentum commutation relations, and the x and z components cannot be measured simultinuously. Thus the numerical values of the Pauli- Lubanski vector should be understood as expectations.
Dec
19
revised Classical vs. Quantum use of the spin 4-vector
added 1 characters in body
Dec
19
answered Classical vs. Quantum use of the spin 4-vector