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Phase Space Was Wholly Cast at That Time, It Was Not Completely Appreciated Until the Late 20Th Century

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Phase Space Was Wholly Cast at That Time, It Was Not Completely Appreciated Until the Late 20Th Century Thomas Curtright, University of Miami WUSTL colloquium 6 December 2017 I will discuss the complete, autonomous formulation of quantum mechanics based on functions that depend simultaneously on the classical variables x and p. Since Heisenberg’s 1927 paper on uncertainty, there has been considerable reluctance to consider positions and momenta jointly in quantum contexts, since these are incom- patible observables. Indeed, Dirac had strong reservations about such an approach. Nev- ertheless, a discomfort to contemplate both positions and momenta simultaneously in the quantum world is not really warranted, as was first fully understood by Hilbrand Groe- newold and José Moyal in the 1940s. While the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely appreciated until the late 20th century. One notable exception was provided by Richard Feynman, who contributed “Negative Probability” to a conference honoring David Bohm in the mid-1980s. In this general talk I will discuss elementary features, as well as some of the early history, of this “deformation” approach to quantization. The subject is of considerable interest in various contemporary contexts. BasedonworkwithDavidFairlieandCosmasZachos. Three alternate paths to quantization: 1. Hilbert space (Heisenberg, Schr¨odinger, Dirac) 2. Path integrals (Dirac, Feynman) 3. Phase-space distribution function of Wigner (Wigner 1932; Groenewold 1946; Moyal 1949; Baker 1958; Fairlie 1964; ...) 1 f(x, p)= dy ψ∗ x − y e−iypψ x + y . 2π 2 2 A special representation of the density matrix (Weyl correspondence). Useful in describing quantum transport/flows in phase space ; quantum optics; quantum chemistry; nuclear physics; study of decoherence (eg, quantum computing). But also signal processing (time-frequency spectrograms); Intriguing mathematical structure of relevance to Lie Algebras, M-theory,... Analysis of distributions on (x, p) phase space involves many techniques that parallel time-frequency analysis, something already familiar to many of you ... so ... Spectrograms pop quiz! A somewhat stylized example ... what is it? 1 https://en.wikipedia.org/wiki/Symphony_No._5_Beethoven An animated spectrograph Fisheye view 2 A less stylized example ... again, what is it? 3 https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.119.161101 4 Partial credit will be given to those who said “black hole merger.” But the time scales are very different. https://www.ligo.caltech.edu/system/media_files/binaries/301/original/detection-science- summary.pdf 5 6 Also known as sonograms for sound ... useful for distinguishing bird calls. “Kay Electric Company started by former Bell Labs engineers Harry Foster and Elmo Crump made a device that was marketed as the ‘Sona-Graph’in 1948. This was adopted by early researchers including C.E.G.Bailey who demonstrated its use for studying bird song in 1950. The use of spectrograms to visualize bird song was then adopted by Donald J. Borror and developed further by others including W. H. Thorpe. These visual representations are also called sonograms or sonagrams. Beginning in 1983, some field guides for birds use sonograms to document the calls and songs of birds. The sonogram is objective, unlike descriptive phrases, but proper interpretation requires experience. Sonograms can also be roughly converted back into sound.” 7 8 9 “ phase space has no meaning in quantum mechanics, thereh being no possibility of assigning numerical values simultaneously to the q’s and p’s ” P. A. M. Dirac See p 132, The Principles of Quantum Mechanics, Oxford University Press, 4th edition, last revised 1967. 2 Properties of f x, p 1 dy ψ∗ x − y e−iypψ x y ( )=2π ( 2 ) ( + 2 ) : Normalized, dpdxf(x, p)=1. Real 2 2 • Bounded: −h ≤ f(x, p) ≤ h (Cauchy-Schwarz Inequality) ; Cannot be a spike: Cannot be certain! • p-orx-projection leads to marginal probability densities: A space- like shadow dp f(x, p)=ρ(x); or else a momentum-space shadow dx f(x, p)=σ(p), resp.; both positive semidefinite. But cannot be conditioned on each other. The uncertainty principle is fighting back ; f can, and most often does, go negative (Wigner). A hallmark of quantum interference. “Negative probability” (Bartlett; Moyal; Feynman; Bracken & Melloy). 1 Some theory and experiments in phase space 4 5 6 7 Oops! f 1 | | ¯h 8 9 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 3 2 0 1 0x -2 -3 An oscillator coherent state (displaced Gaussian) in phase space. 11 Oops again! 12 13 f p x Two-slit experiment in phase space (WF for a state consisting of two Gaussians) 14 16 Nature 386 (1997) pp 150 - 153. 17 1 1 2 Formalism There are three mathematically equivalent but autonomous formulations of quan- tum mechanics based on: 1. Hilbert space –Born,Dirac, Heisenberg, Jordan, Schrödinger 2. path integrals – Dirac,Feynman 3. phase-space – Groenewold, Moyal, von Neumann, Weyl, Wigner While these are equivalent mathematically, they are not always equivalent psycho- logically (as my thesis advisor used to say). One formulation may offer much more insight and provide an easier route to the solution of a particular problem than the others. 2 Wigner Functions Wigner’s original deInition of his eponymous function was (Eqn(5) in [19]) n 1 P (x1, ,xn; p1, ,pn)= dy1 dyn (x1 + y1 xn + yn) ··· ··· ··· ··· ··· 2i(p1y1+ +pnyn)/ (x1 y1 xn yn) e ··· . ··· So deIned, Wigner functions (WFs) reside in phase space. WFs are “Weyl correspondents” of von Neumann’s density operators, . Thus, in terms of Hilbert space position and momentum operators X and P ,wehave1 n n n n n = d d d xd pP (x1, ,xn; p1, ,pn)exp(i (P p)+i (X x)) . (2)2n ··· ··· · · 1 Remark on units: As deIned by Wigner, WFs have units of 1/n in 2n-dimensional phase space. Since it is customary for the density operator to have no units, a compensat- ing factor of n is required in the Weyl correspondence relating WFs to s. Issues about units are most easily dealt with if one works in “action-balanced” x and p variables, whose units are [x]=[p]=. 3 In one x and one p dimension, denoting the WF by f (x, p) instead of P (in deference to the momentum operator P ), we have 1 2ipy/ f (x, p)= dy x + y x y e , | | x + y x y = dp f (x, p) e2ipy/ , | | =2 dxdy dp x + y f (x, p) e2ipy/ x y . | | HWeyl For a quantum mechanical “pure state” 1 2ipy/ f (x, p)= dy (x + y) (x y) e , 2ipy/ (x + y) (x y)= dp f (x, p) e , = , | | where as usual, (x + y)= x + y , x y = (x y). | | EWigner 4 Direct application of the Cauchy—Bunyakovsky—Schwarz inequality to the Irst of these pure-state relations gives 1 f (x, p) dy (y) 2 . | | | | So, for normalized states with dy (y) 2 =1,wehavethebounds | | 1 1 f (x, p) . Such normalized states therefore cannot give probability spikes (e.g. Dirac deltas) without taking the classical limit 0. The corresponding bound in 2n phase-space dimensions is given by the same argument applied to Wigner’s 1 n original deInition: P (x1, ,xn; p1, ,pn) . | ··· ··· | 5 Star Product The star product is the Weyl correspondent of the Hilbert space operator product, and was developed through the work of many over a number of years: H Weyl (1927), J von Neumann (1931), E Wigner (1932), H Groenewold (1946), J Moyal (1949), and G Baker (1958), as well as more recent work to construct the product on general manifolds (reprinted in [20], along with related papers). There are useful integral and diQerential realizations of the product. The integral form is dx1dp1 dx2dp2 i f g = f (x + x1,p+ p1) g (x + x2,p+ p2)exp (x1p2 x2p1) , 2 ( /2) 2 ( /2) /2 x1p2 x2p1 = Area (1,2 parallelogram) , /2=Planck Area = min (xp) , while the diQerential form is i i f g = f (x, p)exp x p p x g (x, p) , 2 2 6 1 1 f g = f x + i p ,p i x g (x, p) 2 2 1 1 = f (x, p) g x i p ,p+ i x 2 2 1 1 = f x + i p ,p g x i p ,p 2 2 1 1 = f x, p i x g x, p + i x . 2 2 The Moyal bracket, f,g = 1 [f,g] , is essentially just the antisymmetric i part of a star product,{{ where}} [f,g] defn.= f g g f. This provides a homomorphism with commutators of operators, e.g. [x, p] = i. 7 Exercises Exercise 1 Non-commutativity. ax+bp Ax+Bp (a+A)x+(b+B)p (aB bA)i/2 e e = e e = Ax+Bp ax+bp (a+A)x+(b+B)p (Ab Ba)i/2 e e = e e Exercise 2 Associativity. eax+bp eAx+Bp ex+p (a+A+)x+(b+B+)p (aB bA+a b+A B)i/2 = e e = eax+bp eAx+Bp ex+p Exercise 3 Trace properties. (a.k.a. “Lone Star Lemma”) dxdp f g = dxdp f g = dxdp g f = dxdp g f Exercise 4 Gaussians. For a, b 0, a b 1 a + b exp x2 + p2 exp x2 + p2 = exp x2 + p2 . 1+ab (1 + ab) 9 Pure States and Star Products Pure-state Wigner functions must obey a projection condition. If the normalization is set to the standard value + dxdp f (x, p)=1, then the function corresponds to a pure state if and only if f =(2) f f. These statements correspond to the pure-state density operator conditions: Tr()=1and = , respectively. If both of the above are true, then f describes an allowable pure state for a quantized system.
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