# Questions tagged [phase-space]

A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

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### Phase space and uncertainty

In phase space every point represents both position and momentum. Isn't it against of Uncertainty principle?
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### Canonical / invertible transformation

What distinguishes a canonical transformation from an ordinary invertible Coordinate transformation? I understand what canonical is but searching the difference between canonical and invertible didn't ...
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### How do I calculate the Altarelli-Parisi splitting function for $g \rightarrow q\bar{q}$?

I'm trying to calculate the Altarelli-Parisi splitting function in the collinear limit for a gluon splitting into a quark-antiquark pair, but I keep getting stuck. Let $p$ and $k$ be the momenta for ...
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### Can every canonical transformation be broken down into a large number of infinitesimal canonical transformations?

Consider a canonical transformation from $(q,p)$ to $(Q,P)$ depending upon a continuous parameter $\alpha$ such that: $$Q_i=Q_i(q,p,t,\alpha), \space P_i=P_i(q,p,t,\alpha)$$ where $q$ and $p$ ...
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### What does integrating the probability density function over all phase space gives us?

For a system of N-3D particles, we have 6N D.O.F and therefore a 6N dimensional phase space. I know that one point in phase space represents a possible state of the system. I also understand that a ...
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### Phase space, ensemble of systems and probability density function

I am trying to understand the concept of phase space in statistical mechanics. I can understand that a system, with $N$ total degrees of freedom, can be in different states, which correspond to ...
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### Operator ordering (for the Groenewold-van Hove theorem)

Suppose we have the (un)usual Schrödinger representation $\pi'(\cdot)$ of the Heisenberg algebra, along with the extension in $\mathbb{sl}(2,\mathbb{R})$ for the quadratic polynomials. It is assumed ...
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### Does the integral of a Wigner function over a finite region mean anything?

I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the ...
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### Classic analog of quantum mechanics when dealing with Hamiltonian operator

I am reading The Principles of Quantum Mechanics by Dirac, in chapter 28 Heisenberg's form for the equations of motion, there is ...
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### Why do Action-Angle Variables form an invariant Torus?

I've been casually reading up on Hamiltonian Mechanics and integrable systems and one term that is used a lot of "invariant torus" where bounded orbits live. KAM theory is also mentioned as ...
Given a phase-space distribution function in special relativity, we can define the following Lorentz invariant vector - $$S^a(x^i) = c\int \frac{d^3p}{E_{p}}P^a f(x^i, p).$$ The spatial component of ...