Quantum-interference effects in some fields

Quantum-interference effects in some fields K. D. KRORI,P. BORGOHAIN, MADHUMITA BARUA,AND KANIKA DAS Mathematical Physics Forum, Cotton College, Gauhati 781001, India Received July 20, 1989

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In this paper, we derive quantum-interference effects for (a) nonspinning particles in Einstein and Yang-Mills fields and (b) spinning particles in Einstein and Einstein-Cartan fields. Nous calculons dans cet art~cleles effets d'lnterfirence quantique pour (a) des partlcules sans spln ddns d-es champs d'Elnsteln et de Yang-M~lls, (b) des part~culesi spln dans des champs d'Elnste~net d'Elnste~n-Cartan [Tradu~tpar la revue] Can J P h y ~68 469 (1990)

1. Introduction In this paper, we derive quantum-interference effects for (a) nonspinning particles in Einstein and Yang-Mills fields, following the treatment of Utiyama (1) and the idea of the Bohm-Aharonov effect (2), and (b) spinning particles In Einstein and Einstein-Cartan fields, following the formalism of Parthasarathy, Rajasekharan, and Vasudevan (PRV) (3). In Sect. 2, we work out a unified gauge-theoretic treatment of Maxwell, Einstein, and Yang-Mills fields following Utiyama; and in Sect. 3, we derive the results for nonspinning particles. We present a brief o u t l ~ n eof the PRV formalism in Sect. 4, and derive the results of spinning particles in Sect. 5, following the treatments of Papapetrou (4) and Hehl (5). 2. Unified gauge-theoretic treatment Let us consider a set of fields ( a = number of particles In a multiplet corresponding to the Yang-Mills field, and for each value of a , A = 1 , 2 , . . . ,N ) with the Lagrangian L(ya", @"). Let this Lagrangian be invariant under a group of transformation G given by

where i" are infinitesimal parameters and ';blB are constant coefficients with a = 1, 2 , 3 , . . . ,n. The coefficients T ; , ~have the properties

we introduce the fields El, (electromagnetic), B; (Yang-Mills), and A: (gravitational) in the form of gauge fields. These fields are treated in a four-dimensional curved space-time. To achieve this, we make use of the functions

where the Latin and Greek indices represent quantities defined with respect to the local Lorentz frame and curved space-time, respectively. Under the transformation group GI, the various field variables transform according to the following scheme:

P.81

FA;

[2.9]

6h; = ikh'P

= i2.A;'

+ r!AP

aik'

+auP

'

Here, fib are called structure constants having the properties The total Lagrangian of the combined field is given by fib =

-fh

Now let the transformation group G be replaced by a wider group GI derived by replacing the parameters 6''s by a set of arbitrary functions ~ ( u ) ' .To make the system still invariant, Pnnted ~nCanada I lmpnme au Canada

+ ~ ~ ( h+ khi^;,) ~ ~ ~+ L)~ ( ~ ; F F ~ ) (see Appendix A for expressions of Fpv, etc.). The variation of the total Lagrangian is given by

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470

CAN. I.

PHYS. VOL. 68,

1990

(See Appendix B for aLT/ayd, etc.). Now let us choose the arbitrary function ~ " ( u )in such a way that e's and integration domain Q. Then the integration of [2.12] over the domain Q gives

&/all

vanish on the boundary surface o f the

with

Since E'S can be chosen arbitrarily within Q, K must vanish at every point in Q; hence the identity [2.12] can be separated into the following two relations:

[2.14]

K

=0

and

Equations [2.14] and [2.15] lead to the following relations:

aLT

+- aL,

[2.22]

-

[2.23]

-

aEv.,

~ L T ~ L T

dB;., [2.24]

=0

a,.,

+-

3q.v

~ L T

-hLp-

3%. ,v

~ L T h,, = 0 ah;. pv

-

From [2.12], the field equations can be written as follows:

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and

=0

ah (LM) = 0 a*e

[2.25]

a L ~-

[2.26]

--

[2.27]

--

12.281

--

[2.29]

--

3wd

~ L M

awd*

4. The Parthasarathy-Rajasekharan-Vasudevan formalism As pointed out by PRV, the change in the phase of the wave function of a particle around a closed path is due to the change in its momentum, and only the parallel displacement of the momentum vector contributes to the change in the momentum around a closed loop. Accordingly, they have written down the change in phase of a particle going around a closed path in five-dimensional Kaluza-Klein theory:

~ L E

aE,

~ L B aB;

a* (a$;3 =)o

~ L T ah:,

where fed is the Christoffel expression in five-dimensional ~ a f u z a - ~ l e itheory. n Then, after some calculation, they have derived the following result:

3. Quantum-interference effects for non-spinning particles in Einstein and Yang-Mills fields The change in phase of a quantum-mechanical wave function for a particle in an electromagnetic field around a closed path is well known (2). This is given by

Now [2.10] can be written in the form

14.21

6W =

jd*,,

(S)

1

S

T$PP&%

9 C

j

A, d r p

where the first term of the right-hand side is the contribution from the Einstein field and the second term is from the Maxwell field, TF* is the usual Christoffel symbol in fourdimensional theory. It may be noted that the result of PRV, [4.2], may also derived within the ambit of four-dimensional theory. We know that the equation of motion of a charged particle in an EinsteinMaxwell field is

where

where Here we put 13.41

T&m=~[Y~,b~

y's being Dirac matrices. By analogy with [3.1], from [3.2] we can write the combined phase shift due to the three fields as follows:

j

Cornparing [4.3] with [4.1] we find that corresponding in the five-dimensional Kaluza-Klein theory, we to T!.rjP have P?E Pa+(il/c)Ft in the four-dimensional theory. The result [4.2ymay be obtalned by integrating this four-dimensional expression over S'.

+ Y,, + G,,) d r p

5. Quantum-interference effects for spinning particles in Einstein and Einstein-Cartan fields

Hence, the phase shifts due to the Yang-Mills and Einstein fields are given by

5.1. Einsteinfield The covariant equation of motion of a spinning particle in

13.51

AW =

(E,

47 2

CAN. J. PHYS. VOL. 68, 1990

general relativity has been given by Papapetrou (4) in the form

where Sap is the spin tensor of the particle, R&,, is the curvature tensor,

where the first term is the general-relativity effect, the second term is the torsion effect, and the third term is the spinning effect of the test particle. If, however, the spin of the particle is zero, then only the first term; i.e., the general-relativity effect, survives.

6. Concluding remarks The importance of the calculations presented in Sects. 2 and 3 lies in showing that Bohm-Aharonov-like quantum interference effects in Yang-Mills and Einstein fields follow immediately from a unified gauge-theoretic treatment. Quantum interference effects in Yang-Mills fields have, perhaps, not been studied experimentally yet. However, an experiment that provides information on the influence of gravity on quantum interference was suggested on the basis of the classical theory of gravitation by Overhauser and Collella (7) and was subsequently performed by Colella et a/. (8). The exact expression of the effect on the basis of the relativistic theory of gravitation is obtained from the PRV formalism (3) outlined in Sect. 4. Now, neutron beams are used to study the quantum-interference effect in the gravitational field of the earth (7, 8); but neutrons have spin. Hence, it is theoretically important to derive the quantum-interference effect for spinning particles. This has been done in Sect. 5. In the context of the growing technological advancement at the present time, it may be possible in the near future to detect the effect due to spin. However, as pointed out by Misner et al. (9), the spin effect would be appreciable in the powerful field of a neutron star or a black hole!

-

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and

Now writing niua

-

P a , we have from [5.1] and [5.2]

Now, extending the PRV technique (as we have done in the case of the Einstein-Maxwell field) to this case, one can see :hat a change in the phase of a spinning particle going around a closed path in the Einstein field may be written down in the form

,(

D +DS

dsp dS (s')

E) DS'

where the first term is the general relativity effect and the second and third terms constitute the spinning effect of the particle.

5.2. Einstein-Car-tan field The Einstein-Cartan theory uses an asymmetric connection of the form

where { ) is Christoffel's symbol and K is the contortion tensor. Hehl ( 5 ) has derived the following equation of motion of a spinning particle in the Einstein-Cartan field:

where SPY is the spin angular momentum of the test particle. Now, again extending the PRV formalism, one can see that the change in the phase of a spinning particle going around a closed path in the Einstein-Cartan field may be written as

Acknowledgements The authors express their profound gratitude to the Government of Assam, Dispur, for all facilities proved at Cotton College, Gauhati 781001, India, to carry out the investigations reported in this paper. 1. R. UTIYAMA. Phys Rev. 101, 1597 (1956). and D. BOHM, Phys Rev. 115, 485 (1959). 2. Y. AHARONOV 3. R. PARTHASARATHY, G. RAJASEKHARAN, and R. VASUDEVAN. Classical Quantum Gravity, 3, 425 (1986). 4. A. PAPAPETROU, Proc. R. Soc. London A, 209, 248 (1951). 5. F. W. HEHL.Phys Lett. 36A, 248 (1951). 6. R. C. TOI-MAN, Relativity, thermodynamics and cosmology. Oxford University Press, Oxford. 1934. and R. COLELLA. Phys Rev. Lett. 33, 1237 7. A. W. OVERHAUSER (1974). 8. R. COLELLA, A. W. OVERHAUSER, and S. A . WERNER. Phys Rev. Lett., 34, 1472 (1975). K. S. THORNE, and J. A. WHEELER. Gravitation. 9. C. W. MISNER, W.H. Freeman and Company, San Francisco. 1973.

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KRORl ET AL.

Appendix B 473