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Time Reversal Symmetry and Humpty-Dumpty * Ber thold-Georg Englert Center for Advanced Studies and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, 87131 and Sektion Physik, Universität München, Am Coulombwall 1, D-85748 Garching

Z. Naturforsch. 52a, 13-14 (1997)

It is argued that the quantal evolution of a physical system is fundamentally irreversible although the dynamical equations are symmetric under time reversal. The example of spin coherence in a Stern-Gerlach interferometer demonstrates that even undoing the evolution partially only may require sub-microscopic precision in controlling a macroscopic apparatus.

1. Elementary Considerations

In simplest nonrelativistic quan tum mechanics - a particle with mass m moving along the x-axis under the influence of a potential energy V (x) - the dynam-ics is governed by the Hamil ton opera tor

H = ~ p 2 + K ( x ) > 0 , 2m

where, for simplicity, the requirement that H be bounded f rom below is equivalently replaced by in-sisting on the positivity of H. The corresponding Schrödinger differential opera tor Q,

h 2 0 2

h Q = - — - - j + V(x), 2 m ox

appears in the Schrödinger equat ion for the spatial probabi l i ty ampl i tude iJ/ (t, x),

ER + iü )\j/(t, x) = 0 .

It is well known an d easily demonst ra ted that the time reversed wave funct ion ( 0 T iJ/) {t, x) = ij/*{2T — t, x) is also a solution to this Schrödinger equation,

— +iQ ) iA*(2T - r , x ) = 0

* Presented at a Workshop in honor of E. C. G. Sudarshan's contributions to Theoretical Physics, held at the Univer-sity of Texas in Austin, September 15-17, 1991.

Reprint requests to Dr. B.-G. Englert, now at Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching.

The ant i -uni tary time reversal opera to r 0 T has these effects on spatial and momenta l wave functions:

( 0 r ^ ) ( T , x) = \J/* (T, x ) ,

(0riJ/)(T,p) = il/*(T,-p).

Perhaps most t ransparent is the t ransformat ion of the Wigner funct ion gw(t, x, p) associated with the state of the system: {0TqJ(T, X, p) = qw(T, x, - p), signify-ing that the momen tum, here identical with the veloc-ity, is reversed. Indeed, for more complicated dynam-ics it is the velocities, not canonical momenta , that are reversed.

2. Undoing Previous Evolution

With the time reversal opera tor at hand one can imagine the undoing of all evolut ionary changes that have occurred between, say, t = 0 and t = T by first applying 0 T , followed by evolution under the same Hamil ton opera to r for T < t < 2 T, and final applica-tion of 0 2 X , so that an instant x after t = 2 T one has the same wave funct ion as at time t = z:

x\j{2T + T, x) = e-iQz02Te~iQz0Te-iQz\lj{<d, x)

= \Mt, x) = e~iQx\l/{0, x ) .

This, however, cannot be achieved in any real sense. The changes symbolized by 0T and 0 2 t must be of dynamical origin themselves, which they cannot be because these opera tors are not unitary. It is true that all that really mat ters is the produc t

02Te-iQT 0T = eiQT

and this is unitary. But for most Q the r ight-hand side is not of the form exp (— i Q T) with a physical, posi-

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14 B.-G. Englert • Time Reversal Symmetry and Humpty-Dumpty

tive Q. We are thus led to the conclusion that the undoing of all evolut ionary changes is impossible. In other words: Q u a n t u m mechanical evolution is funda-mentally irreversible.

3. The Humpty-Dumpty Problem

A more general scheme, not so tightly bound to elementary nonrelativistic q u a n t u m mechanics is what I would like to call the Humpty-Dumpty prob-lem* Given an initial state with density opera tor (?(0) and a Hamil ton o p e r a t o r H l (t) > 0, acting from t = 0 to t = T, find T and H2 (t) > 0, acting f rom t = T to t = T, such that the unitary evolution opera tors (7(0, T) and U(T, T), constructed as time ordered exponentials f rom Hx (t) and H2{t), respectively, are effectively equal to the unit opera tor

1/(0, T)U(T, T) = 1

when applied to the given p(0). The realization re-quires

exp + -\H2(t)dt ft t

exp - J t f i W d r n o

where we have s tandard time order ing on the right and the reverse order on the left. In view of the positiv-ity of both H x (t) and H2(t) this can only by realized under exceptional circumstances. The H u m p t y -Dumpty problem is then: F o r which £?(0) and H j (r) is it possible at all? There are, of course, two very impor-tant rules in this game, namely (i) only real physical interactions are considered, and (ii) over-idealizations are not allowed.

4. Spin Coherence in a Stern-Gerlach Interferometer

Maybe part of the evolut ionary changes can be un-done at least? Consistent with rule (i) we ask more specifically: Can the two partial beams of a Stern-Ger-

* If you don't have a copy of Mother Goose at home, you can look up the Humpty-Dumpty riddle in [2] and [3].

lach appara tus (SGA) be reunited with such precision that the initial spin state is recovered? This problem of construct ing a SG interferometer has been studied some time ago by Schwinger, Scully, and myself [1, 2]; a gedanken experiment based on a funct ioning device was also analyzed [3], It is found that one needs three more SGAs for the beam reunion. Ideally, the four SGAs would be perfectly identical. With due respect to rule (ii), however, we have to ask how large a mis-match between the SGAs can be tolerated. Two rele-vant quanti t ies are the net transverse m o m e n t u m transferred to either partial beam and their net trans-verse displacement, measured by

Ap = \dtF(t), Az = - jdtF(t)t/m,

where F(t) is the force on the up-component , for in-stance, produced by the inhomogeneity of the mag-netic field. Thus, Ap and Az are propert ies of the macroscopic SGAs; the deviations f rom the ideal values Ap = 0, Az = 0 are resulting f rom the lack of control over the SGAs. It turns out that to mainta in spin coherence one must, at least, ensure that

(5z Ap/h)2 + (bp Az/h)2 <§ 1

holds, where 5z and 5p are the natural spreads of the beam prior to entering the first SGA. Therefore, one can tolerate only such variations in the magnet ic field that bo th | Az | h/bp and | Ap | h/bz, with the con-sequence

| Az 11Ap| h

h bz bp

In summary, this states that the macroscopic magnet ic field must be controlled with sub-microscopic preci-sion. It won't be easy.

Acknowledgement

It is a great pleasure to dedicate these remarks to E. C. G. Sudarshan whose interest in irreversible quan-tal evolution is on record [4], I am grateful for the valuable insights gained in discussions with the co-authors of the H u m p t y - D u m p t y papers [1-3] , J. Schwinger and M . O . Scully, as well as with G. Süßmann and I. Bialynicki-Birula. This work was par-tially supported by the Office of Naval Research.

[1] B.-G. Englert, J. Schwinger, and M. O. Scully, Found. Phys. 18,1045 (1988).

[2] J. Schwinger, M. O. Scully, and B.-G. Englert, Z. Phys. D 10, 135 (1988).

[3] M. O. Scully, B.-G. Englert, and J. Schwinger, Phys. Rev. A 40, 1775 (1989).

[4] E. C. G. Sudarshan, C. B. Chiu, and V. Gorini, Phys. Rev. D 18, 2914 (1978).