12,213 reputation
21233
bio website
location
age
visits member for 3 years, 5 months
seen 37 mins ago

Mar
1
comment Is there a Lagrangian formulation of statistical mechanics?
@Nathaniel cont. In this case instead of an iterative procedure of guessing the initial momenta and checking the second end boundary condition, we can solve the Hamilton-Jacobi equation. The price is that it is a partial differential equation. A second applications is in the discrete formulation of mechanics (we can imagine that the time "t" to be a small time step). I haven't seen an application to statistical mechanics. This is why I thought that the question is very original. But I haven't searched hard enough.
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.
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
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
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
11
comment Beyond WKB approximation for energies
@Emilio You may find these lecture note by B.C. Hall useful: math.cinvestav.mx/~NorteSur/Taller10/notas/Hall.pdf
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
If I may add, if you write $a = x+ip$ and $a^{\dagger} = x-ip$, you get the harmonic oscillator Hamiltonian in the usual representation, but the harmonic osillator coordinates $x$ and $p$ are not the original coordinates that you started with.
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
If you substitute the expressions of $a$ and $a^{\dagger}$ given in the answer in the Hamiltonian $H=\hbar\omega(a a^{\dagger}+\frac{1}{2})$ you get the Hamiltonian you started with expressed in terms of $X$ and $P$.
Nov
20
comment Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
The Hamiltonian expressed in terms of the creation and annihilation operators has exactly the form of the harmonic oscillator Hamiltonian. Thus the energy levels are exactly equal to those of the harmonic oscillator, however with infinite degeneracy per level (The harmonic oscillator energy levels are nondegenerate). The advantage of using this method is that it allows an algebraic solution of the energy levels (i.e. without solving differential equations), please see the quantum harmonic oscillator Wikipedia page: en.wikipedia.org/wiki/Quantum_harmonic_oscillator
Nov
19
comment What are the solution spaces of Nonlinear Schrödinger equations?
@Arnold The following work by Forger and Romero establishes the equivalence betwee Crnkovic-Witten-Zuckerman and the Peierls brackets and relates them to the multisymplectic approach arxiv.org/pdf/math-ph/0408008.pdf
Nov
19
comment Interacting representation of the Poincaré group
cont., In 4D scalar theory, in the free case, the (free) Poincaré generators can be expressed in terms the field's creation and anihilation operators and their action on the corresponding (free) Fock space is completely known. In addition approximate representations can be constructed order by order in perturbation theory in which the interacting Poincaré generators will have higher polynomial dependence on the creation and anihilation operators (acting on the free Fock space), but an exact representation is not known.
Nov
19
comment Interacting representation of the Poincaré group
@BGal One can easily find an interaction term, and construct the expressions of the interacting algebra in terms of the field variables. For example in a scalar field theory \mathcal{H} = \lambda \phi^4 is a possible interaction term, and it is easy to exactly construct interacting Poincaré generators in terms of the field and its derivatives. Finding a representation on the other hand would mean to find a Hilbert space on which the action of the interacting Poincaré generators is completely known.
Nov
18
comment What are the solution spaces of Nonlinear Schrödinger equations?
@Arnold, please see the Crncovic-Witten and Zuckerman's articles given in Urs Schreiber's answer physics.stackexchange.com/questions/26883/…. The Crncovic-Witten's link is not working, but you can find their article in the book: books.google.co.il/…
Nov
18
comment What are the solution spaces of Nonlinear Schrödinger equations?
@Arnold, Please see for example fiz.uni.opole.pl/pgar/documents/IJMPA87.pdf by Piotr Garbaczewski
Nov
6
comment Symmetries of spacetime and objects over it
@Qmechanic Thank you
Oct
16
comment Mathematically challenging areas in Quantum information theory and quantum cryptography
Coherent states which are widely used in quantum cryptography, have a geometric origin (Bargmann space). Please see for example Ma's thesis: web.williams.edu/go/math/sjmiller/public_html/crypto/handouts/…
Oct
4
comment How to construct the charge conjugation matrix for any given dimension?
I can show you how to construct in this particular case, but can you please check your calculations because in your expressions $(\Gamma^1)^2=(\Gamma^2)^2=1$ while $(\Gamma^3)^2=(\Gamma^4)^2=-1$, i.e., you are working in a signature $(1, 1, -1, -1)$. Is this really the case you need.