Thomas Curtright, University of Miami KU Lawrence Colloquium 30 April 2012 Since Heisenberg's 1927 Paper on Uncertainty, There

Thomas Curtright, University of Miami KU Lawrence colloquium 30 April 2012

Since Heisenberg’s 1927 paper on uncertainty, there has been considerable reluc- tance to consider positions and momenta jointly in quantum contexts, since these are incompatible observables. But a discomfort to contemplate both positions and momenta simultaneously in the quantum world is not really warranted, as was Hrst fully understood by Hilbrand Groenewold 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. In this general talk I will discuss elementary features, as well as some of the early history, of this “deformation” approach to quantization.

Based on work with David Fairlie, Andrzej Veitia, and Cosmas Zachos.

1 “ 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 However, please also see:

and references therein, as well as the Asia-Paci#c Physics Newsletter, May 2012 (premier issue).

3 1 Some theory and experiments in phase space

4 5 6 7 Oops! f 1 | | ¯h

8 9 0.3

0.2

0.1

-0.1

-0.2

-0.3

3 2 0 1 0x -2 -3

Wigner function for the oscillator state 1 ( 0 + i 1 ). 2 | |

10 0.3

0.2

0.1

-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 f

Two-slit experiment in phase space (WF for a state consisting of two Gaussians)

15 16 Nature 386 (1997) pp 150 - 153.

17 Formalism

There are three mathematically equivalent but autonomous formulations of quantum mechanics based on:

1. Hilbert space

–Born,Dirac, Heisenberg, Jordan, Schrödinger

2. path integrals

– Dirac,Feynman

3. phase-space

–anti-Dirac, Groenewold, Moyal, von Neumann, Weyl, Wigner

While these are equivalent mathematically, they are not always equivalent psychologically (as my thesis advisor used to say). One formulation may oQer 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 , 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 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. Otherwise not. You can easily satisy only one out of these two conditions, but not the other, using an f that is not a pure state.

Without drawing on the Hilbert space formulation, it may at Irst seem to be rather remarkable that explicit WFs actually satisfy the projection condition (cf. the above Gaussian example, for the only situation where it works, a = b =1, i.e. exp ( (x2 + p2) /)). However, if f is known to be a eigenfunction with non-vanishing eigenvalue of some phase-space function with a non-degenerate spectrum of eigenvalues, then it must be true that f f f as a consequence of associativity, since both f and f f would yield the same eigenvalue.

8 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 The Simple Harmonic Oscillator

There is no need to deal with wave functions or Hilbert space states. The WFs may be constructed directly on the phase space [9, 2]. Energy eigenstates are obtained as (real) solutions of the -genvalue equations [6]:

H f = Ef = f H. To illustrate this, consider the simple harmonic oscillator (SHO) with (m =1, =1) 1 H = p2 + x2 . 2 The above equations are now second-order partial diQerential equations,

1 1 2 1 2 H f = p i x + x + i p f = Ef , 2 2 2 1 1 2 1 2 f H = p + i x + x i p f = Ef . 2 2 2 But if we subtract (or take the imaginary part),

2 2 (p x x p) f =0 f (x, p)=f x + p . So H f = Ef = f H reducestoasingleordinary diQerential equation (Laguerre, not Hermite!), namely, the real part of either of the previous second-order equations.

10 There are integrable solutions if and only if E =(n +1/2) , n =0, 1, for which ···

n 2 2 ( 1) x + p (x2+p2)/ fn (x, p)= Ln e , /2 n 1 z d n z L (z)= e z e . n n! dzn

+ The normalization is chosen to be the standard one dxdp fn (x, p)=1. Except for the n =0ground state (Gaussian) WF, these f’s change sign on 1 2 the xp-plane. For example: L0 (z)=1,L1 (z)=1 z, L2 (z)=1 2z + 2 z , etc. Using the integral form of the product, it is now easy to check these pure states are orthogonal:

(2) fn fk = nk fn . This becomes more transparent by using raising/lowering operations to write2

1 n n f = (a ) f ( a) n n! 0 1 n (x2+p2)/ n = (a ) e ( a) , n! 1 where a is the usual linear combination a (x + ip), and a is just 2 1 its complex conjugate a (x ip) ,witha a a a =1, and 2 a f0 =0=f0 a (cf. coherent state density operators).

2 Note that the earlier exercise giving the star composition law of Gaussians immediately yields the projection property of the SHO ground state WF, F0 =(2) F0 F0.

11 0.2 0.4 0.1 0.3

0.2 0.0

F(x,p) 0.1 F(x,p) -0 .1

0.0 -0 .2

-0 .1 -0 .3

-2 -2

0 -2 0 -2 p 0 p 2 x 0 2 2 x 2

n =0 n =1

0.2 0.3 0.1 0.2 0.0 0.1 F(x,p) F(x,p) -0 .1 0.0 -0 .2 -0 .1 -0 .3 -0 .2

-2 -2

0 -2 0 -2 p p 0 0 2 x 2 x 2 2

n =2 n =3

12 The Uncertainty Principle

Expectation values of all phase-space functions, say G (x, p),forasys- tem described by f (x, p) (a real WF, normalized to 1) are just integrals of ordinary products (cf. Lone Star Lemma)

G = dxdp G (x, p) f (x, p) . These can be negative, even though G is positive, if the WF Jips sign. So how do we directly establish simple correct statements such as (x + p)2 0 without using marginal probabilities or invoking Hilbert space results? The roles of positive-deInite Hilbert space operators are played on phase space by real star-squares:

G (x, p)=g (x, p) g (x, p) .

These always have non-negative expectation values for any g and any WF,

g g 0 , even if f becomes negative. (Note this is not true if the is removed! If 2 g g is supplanted by g , then integrated with a WF, the result could be negative.) | |

13 To show this, Irst suppose the system is in a pure state. Then use f =(2) f f (see Pure States and Star Products), and the associativity and trace properties (see Exercises), to write:3

dxdp (g g) f =(2) dxdp (g g)(f f) =(2) dxdp (g g) (f f) =(2) dxdp (g g f) f =(2) dxdp f (g g f) =(2) dxdp (f g) (g f) =(2) dxdp (f g)(g f) 2 =(2) dxdp g f | | 0 .

In the next to last step we also used the elementary property (g f) = f g.

3 By essentially the same argument, if F1 and F2 are two distinct pure state WFs, the phase-space overlap integral between the two is also manifestly non-negative and thus admits interpretation as the transition probability between the respective states: 2 2 dxdp F1F2 =(2) dxdp F1 F2 0 . | |

14 More generally, if the system is in a mixed state, as deIned by a normalized probabilistic sum of pure states, f = jfj, with probabilities j 0 j P P satisfying j =1, then the same inequality holds. j P Correlations of observables follow conventionally from speciIcchoicesof g(x, p). For example, to produce Heisenberg’s uncertainty relation, take

g (x, p)=a + bx + cp , for arbitrary complex coePcients a, b, c. The resulting positive semi-deInite quadratic form is then

1 x p a g g = a b c x x x x p b 0 , pp xp p c for any a, b, c. All eigenvalues of the above 3 3 hermitian matrix are therefore non-negative, and thus so is its determinant,×

1 x p det x x x x p 0 . pp xp p

15 But

2 2 x x = x ,p p = p ,x p = xp + i/2 ,p x = xp i/2 , andwiththeusualdeInitions of the variances

(x)2 (x x )2 , (p)2 (p p )2 , the positivity condition on the above determinant amounts to

2 2 1 2 2 (x) (p) +( xp x p ) . 4 Hence Heisenberg’s relation

(x)(p) /2 . The inequality is saturated for a vanishing original integrand g f =0,for suitable a, b, c,whenthe xp x p term vanishes (i.e. x and p statistically 1 independent, as happens for a Gaussian pure state, f = exp ( (x2 + p2) /)).WHeisenberg

16 A Classical Limit

The simplest illustration of the classical limit, by far, is provided by the SHO ground state WF. In the limit 0 a completely localized phase- space distribution is obtained, namely, a Dirac delta at the origin of the phase space: 1 2 2 lim exp x + p / = (x) (p) . 0 Moreover, if the ground state Gaussian is uniformly displaced from the origin by an amount (x0,p0) and then allowed to evolve in time, its peak follows a classical trajectory (see Movies ... this simple behavior does not hold for less trivial potentials). The classical limit of this evolving WF is therefore just a Dirac delta whose spike follows the trajectory of a classical point particle moving in the harmonic potential:

1 2 2 lim exp (x x0 cos t p0 sin t) +(p p0 cos t + x0 sin t) / 0 = (x x0 cos t p0 sin t) (p p0 cos t + x0sin t) .

17 2 Some history of QMPS

Zeitschrift für Physik 46 (1927) pp 1-33

18 Mathematische Annalen 104 (1931) pp 570-578.

On p 574:

where “ ” (pp 571, 573)

19 Physical Review 40 (1932) pp 749-759

20 21 Evidently, Wigner was well-aware of von Neumann’s work on density operators:1

= . | | Not too long after Wigner’s paper appeared, von Neumann went to Cambridge, in 1934. His presence there enticed a young student of physics from the Netherlands, “Hip” Groenewold, to also visit Cambridge. Hip was so strongly inIuenced by von Neumann’s lectures and by the ideas expressed in the latter’s book, Mathematische Grundlagen der Quantenmechanik, that he resolved to work on the principles of QM for his doctoral thesis. Groenewold returned to the Netherlands in 1935 and continued his graduate stud- ies. After a decade and a world war, his thesis was Hnished.

1 For that matter, Dirac almost anticipated Wigner’s construction. See Proceedings of the Cambridge Philosophical Society 26 (1930) pp 376-385.

22 Physica XII (1946) pp 405-460

Especially on pages 451 and 452:2

This should be compared with von Neumann’s integral form as given above. The von Neumann and Groenewold expressions for the product are equivalent, as may be seen after performing yet another pair of Fourier transforms.

See p 447 following Eqn (3.14).

23 And as if to stress his last point about the commutator, Groenewold repeats it again a few lines later.

24 Groenewold also explicitly shows the Wigner function is the Weyl correspondent of the von Neumann density operator.

25 P A M Dirac, “On the Analogy Between Classical and Quantum Mechanics” Reviews of Modern Physics 17 (1945) 195-199 – Bohr’s festschrift.

Evidently, at the time (April-June 1945) Dirac was unaware of (or had forgotten about) Wigner’s 1932 paper. Moyal was also unaware of Wigner’s work, initially, but he eventually learned of the 1932 paper and brought it to Dirac’s attention on 21 August 1945.

26 (9 January 1946)

The complete set of letters between Moyal and Dirac is held in the J.E. Moyal Pa- pers, Basser Library, Australian Academy of Science, Canberra, and is reproduced in Maverick Mathematician, by Ann Moyal. Also see the Dirac collection at FSU.

27 Proceedings of the Cambridge Philosophical Society, Vol. 45, 1949, pp 99-124.

Especially on pages 106-108.

28 Although, this particular statement is already in Wigner, at least for the case 1 2 H = 2m p + V (x).

29 However, Moyal then makes a more sweeping statement, which comes out of the blue, in my opinion.

30 Of course, Moyal’s result (7.10) is already in Groenewold [his Eqn’s (4.29) and (4.38) shown above]. Moreover, so far as the last sentence in Moyal’s paragraph above, this is now known as the “Groenewold—van Hove theorem” for which the Hrst explicit counterexample is in Groenewold (see his Eqns (4.12) and (4.13)). The quantum correspondent of the classical Poisson bracket combination 1 x3,p3 + p2,x3 , x2,p3 =0 PB 12 PB PB PB does not vanish when computed in QMPS:

1 1 1 1 x3,p3 + p2,x3 , x2,p3 = 3¯h2 =0 ih¯ 12ih¯ ih¯ ih¯ This is isomorphic to the corresponding Hilbert space statement for operators X and P , and their commutators.

1 1 1 1 X3,P3 + P 2,X3 , X2,P3 = 3¯h2 =0 ih¯ 12ih¯ ih¯ ih¯

31 Finally, in his acknowledgements, Moyal says:

But, sadly, so far as Moyal’s correspondence with Groenewold is concerned, as Ann Moyal states in Maverick Mathematician,p43,footnote28:

“This correspondence has not survived.”

32 As Groenewold’s example illustrates, it is not true that commutators are always given by ih¯ times Poisson brackets, as advocated by the young student (23 !!!)

P. A. M. Dirac, Proc. R. Soc. Lond. A 109 (1925), pp 642-653

In particular, see p 648. After deHning

Dirac said:

33 In retrospect, it would seem Dirac missed an opportunity to correct for this indis- cretion of his youth, by collaborating with Moyal in the development of the exact quantum bracket.

Thank you for the invitation to KU and for the opportunity to present this material.

34 References

[1] Ann Moyal, Maverick Mathematician, ANU E Press (2006) (online)

[2] M Bartlett and J Moyal, “The Exact Transition Probabilities of Quantum-Mechanical Oscillators Calculated by the Phase-Space Method” Proc Camb Phil Soc 45 (1949) 545-553.

[3] P A M Dirac, “Note on Exchange Phenomena in the Thomas Atom” Proc Camb Phil Soc 26 (1930) 376-385.

[4] P A M Dirac, “On the Analogy Between Classical and Quantum Me- chanics” Rev Mod Phys 17 (1945) 195-199.

[5] P A M Dirac (1958) The Principles of Quantum Mechanics, 4th edition, last revised in 1967.

[6] D Fairlie, “The Formulation of Quantum Mechanics in Terms of Phase Space Functions” Proc Camb Phil Soc 60 (1964) 581-586. Also see T Curtright, D Fairlie, and C Zachos, “Features of Time Independent Wigner Functions” Phys Rev D58 (1998) 025002.

18 [7] J Gani, “Obituary: José Enrique Moyal” J Appl Probab 35 (1998) 1012- 1017.

[8] J Gleick, Genius, Pantheon Books (1992) p 258.

[9] H J Groenewold, “On the Principles of Elementary Quantum Mechan- ics” Physica 12 (1946) 405-460.

[10] N Hugenholtz, “Hip Groenewold, 29 juni 1910-23 november 1996” Ned- erlands Tijdschrift voor Natuurkunde 2 (1997) 31.

[11] D Leibfried, T Pfau, and C Monroe, “Shadows and Mirrors: Recon- structing Quantum States of Atom Motion” Physics Today, April 1998, pp 22-28.

[12] J E Moyal, “Quantum Mechanics as a Statistical Theory” Proc Cam- bridge Phil Soc 45 (1949) 99-124.

19 [13] J von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press (1955, 1983).

[14] D Nolte, “The tangled tale of phase space” Physics Today, April 2010, pp 33-38.

[15] J. Preskill, “Battling Decoherence: The Fault-Tolerant Quantum Com- puter” Physics Today, June (1999).

[16] S Saunders, J Barrett, A Kent, and D Wallace, Many Worlds?, Oxford University Press (2010).

[17] R J Szabo, “Quantum Ield theory on noncommutative spaces” Physics Reports 378 (2003) 207-299.

[18] H Weyl, “Quantenmechanik und Gruppentheorie” Z Phys 46 (1927) 1- 33.

[19] E Wigner, “Quantum Corrections for Thermodynamic Equilibrium” Phys Rev 40 (1932) 749-759.

[20] C Zachos, D Fairlie, and T Curtright, Quantum Mechanics in Phase Space, World ScientiIc, 2005. (a revised and updated Irst chapter is online)