# Measurements in the phase space picture of quantum mechanics

Suppose we are dealing with non relativistic quantum mechanics of point particles, that is we are in the realm of 'classic' quantum mechanics (no quantum fields ect.).

In the Heisenberg picture, observables are described by self adjoint operators, that act on a Hilbert space. Given a state and an observable, a measurement is modeled in such a way, that all eigenvalues of that operator are the possible outcomes.

An actual measurement then has one of the eigenvalues as its result and after the measurement the measured state vector is projected (collapsed if you like) onto the appropriate eigenvector of that eigenvalue.

On the other hand, the phase space description of quantum mechanics describes observables as functions on the phase space, but not with the ordinary 'dot' product (as in classical mechanics on the phase space) but with a $\star$-product. For example the Moyal product.

Now both descriptions are exactly equivalent via the Weil-Wigner transformation. That is there is a priori no reason to favor one over the other.

My question is: How is measurement and measurement outcome modeled in the $\star$-product phase space description of QM?

• Hermann Weyl is credited with co-inception of the map, not the distorted-homophone French mathematician. What specific experiment do you have in mind that could possibly depend on one of several equivalent QM formalisms but not others? Many quantum optics experiments are analyzed through both Hilbert space and the phase space formalisms, as standard textbooks detail. – Cosmas Zachos Feb 2 '18 at 14:42
• @SeanE.Lake : operators are instantly mapped into c-number functions by the Wigner map, provided they are written down explicitly and specifically. Reviews in Physics Today, etc., explain how Wigner functions are thus reconstructed ("measured"). But one must first make sure there are no confusions or misgivings about the measurement process; which is the point of the quest for a specific, narrow example. I suspect the OP does not care for slit experiment interference patterns.... – Cosmas Zachos Feb 2 '18 at 15:57
• Any experiment, that serves the purpose of explanation is good enough, I think. Slit experiment or whatever. I'm after the translation of the rule for measurements in the Heisenberg picture into the phase space picture. – Mark Neuhaus Feb 2 '18 at 16:24
• You misspelled Hermann Weyl's name.... (Andre Weil Never dealt with this... Never.) – Cosmas Zachos Feb 2 '18 at 17:05
• Yes, there is no "if" part in the equivalence. Never said that. Don't know what brings you to this conclusion. both pictures ARE equivalent. Therfore the collaps of the state at a measurement MUST have a counterpart in the phase space picture. – Mark Neuhaus Feb 2 '18 at 18:30

Conceptually, little has changed for 20 years to moot this outstanding Physics Today article on the subject.

Typically, in the iconic double-slit interference experiment, wave patterns on a screen are observed and measured, and compared with QM predictions, typically expectation values in position space. Through the strength of mathematical theorem, the predictions of Hilbert-space, or phases-space QM are identical, inevitably.

In phase-space quantization, one does the momentum integral to produce a space probability distribution, identically to that produced in Hilbert space, and the related expectation values derived in standard texts. They are always consistent with the uncertainty principle, at the heart of QM. This principle is straightforward to derive through commutator-based inequalities and the positivity of the norm in the Hilbert space formulation; and through even more intriguing $\star$-product inequalities and subtler mathematical features in the phase-space formulation. (That product was discovered by H Groenewold in 1946---Moyal does not appear to have known about it at the time.)

Edit to address comment . A discontinuous change in the wave function following measurement, is translatable, mutatis mutandis , as a discontinuous collapse/change in the Wigner Function, the celebrated Wigner transform of the density matrix. So, an enhanced "localization" of the wave function corresponds to a "squeezing" of the Wigner function. This Wigner function corresponds to the quasi-probability measure in phase space and is interconvertible to wave functions. It evolves "analogously" to wave functions through the Moyal equation--the analog of the von Neumann equation in Hilbert space. Here is its double-slit profile before travel to the screen and collapse,

So, nothing in the measurement process would present any differently in this formulation.

• What I mean is the following: In the Heisenberg picture a measurement changes the system in a discontinuous way, as it project the state onto an eigenstate of that operator. Thats measurement and the collaps of the wave function. What is the exact counterpart to this discontinuous collaps in the phase space picture via the Weyl-Wigner correspondence? What you described is the correspondence between the measurable values and the mathematical objects which describe the measurements in both pictures. I'm aware of that. What I miss in the phase space picture is the equivalent of the collaps – Mark Neuhaus Feb 2 '18 at 18:19
• Or in other words: How do we know in the phase space picture, if a measurement has taken place? What is different after the measurement. In the Heisenberg picture, its the state that is different. After the measurement its the appropriate eigenstate, regardless of what it was before. – Mark Neuhaus Feb 2 '18 at 18:24