Proc. Nati. Acad. Sci. USA Vol. 80, pp. 7051-7053, November 1983 Physics

Classical geometric resolution of the Einstein-Podolsky- Rosen paradox (quantum mechanics/nonseparability/gauge theory/fiber bundle) YUVAL NE'EMANt Tel Aviv University, Ranat Aviv, 69978, Israel; and Institute for Advanced Study, Princeton, NJ 08540 Contributed by Yuval Ne'eman, July 7, 1983

ABSTRACT I show that, in the geometry of a fiber bundle EPR nonlocal action describing a gauge theory, curvature and parallel transport en- sure and impose nonseparability. The "Einstein-Podolsky-Rosen The typical EPR experiment deals with a composite state (AB paradox" is thus resolved "classically." I conjecture that the os- ...) at the origin 0 that then decomposes, the constituents A, tentatiously "implausible" features of the quantum treatment are B, etc., thus. going their separate ways. The composite state due to the fact that space-time separability, a basic assumption of carries a certain irreducible representation D()mp of the group single-particle nonrelativistic quantum mechanics, does not fit the G, appearing in the Clebsch-Gordan decomposition of the bundle geometry of the complete physics. product DA X DB X ...

Fiber bundle geometry reproduces. nonlocal quantum effects DA®DB *- = E Dn)mp. [1] semiclassically n A measurement of the observable go, a quantum number of Those features of quantum mechanics that appear to disagree G or of its generator algebra g (for a Lie group) performed on with a classically conditioned physical intuition are of two types: the constituent A constrains through Eq. 1 the values of the features relating to the act of measurement (collapse of the state- same quantum number at B, C, etc. vector) and nonlocal features. The latter include the Aharonov- A simple example of an EPR situation is provided by fIb de- -Bohm (1) effect; monopoles both of the Dirac type (2) in elec- cay (20). The two y rays should have their spin polarizations tromagnetism and of the 't Hooft-Polyakov type (3) in Yang- adding up to zero (and to negative total intrinsic parity) when Mills (YM) theories (4, 5); other YM structures such as instan- observed, whatever the AB distance. For example, once the tons (6, 7), merons, and such; and Einstein-Podolsky-Rosen (8) helicity of A is measured and found to be + 1, a measurement (EPR) simulated action at a distance. In recent years, the Ahar- of B will have to yield -1 with certainty. In the contending onov-Bohm (1) effect [first tested experimentally by Chambers realism view, this would be understood classically as indicating (9)] was given a semi-classical geometric realization, as a to- that the physical allocation of helicities had actually preceded pological global effect (10). Electromagnetic gauge invariance both measurements and is part of the underlying reality searched is realized geometrically on a fiber bundle (11, 12). Such a man- for by Einstein and his collaborators. The probabilistic answer ifold given by quantum mechanics is then supposed to represent an (13-15) corresponds to a nontrivial juxtaposition of a (ver- epistemological uncertainty (i.e., lack of knowledge), rather than tical) group space (or group representation space), the fiber at an intrinsic indeterminacy. However, experimental verification each point of the (horizontal) base space (generally space-time). of the existence of a fundamental indeterminacy has been pro- The relevant group constituting the fiber is in this case the phase- vided, e.g., through the falsification of Bell's inequalities (21- invariance group. (This is in fact the only data with a quantum 24), which had indeed assumed the existence of such an un- origin, the rest of the description being entirely classical.) The derlying reality. We are thus forced to accept nonseparability fiber bundle is a locally trivial juxtaposition, yet allowing for a or apparent action at a distance. When one usually adds here nonsimply connected geometry to arise globally as in the case the postulate of special relativity (a somewhat inconsistent pro- of a Mobius strip, where a twist is introduced globally only. In cedure in treating nonrelativistic quantum mechanics, to say Yang's treatment of the Dirac monopole (16), the global effect the least), nonseparability appears to imply in addition acausal is achieved by the removal of a point at the origin of the R3 base behavior, because there is no way for the results of a mea- space. It is this nontrivial structure that creates nonshrinkable surement in A to propagate fast enough so as to affect a (lab- loops or n-spheres and results in discrete topological effects oratory frame) simultaneous helicity measurement in B. (winding numbers) reproducing both the Aharonov-Bohm ef- A point that does not appear to have been generally dis- fect itself (12) and monopoles (16), instantons, merons, and such. cussed in the study of the paradox relates to the local definition In recent years, several papers have reported this geometric of the angular momentum generators themselves (in nonrela- realization of YM gauges (4, 5). tivistic quantum mechanics, this is of course, axiomatic) or of In the present note, I point out that the geometry of fiber the momenta in the original EPR gedanken experiment. How bundles (13-15, 17-19) provides in addition a classical geo- can these algebraic generators be defined in two different places metric realization of EPR nonlocal action or nonseparability of and still be assumed to represent the same entity? How do we the composite wave function. know that the two helicities can still be added to zero? Could

The publication costs of this article were defrayed in part by page charge Abbreviations: EPR, Einstein-Podolsky-Rosen; YM, Yang-Mills. payment. This article must therefore be hereby marked "advertise- t Also Ministry of Science and Development, Jerusalem, Israel; and on ment" in accordance with 18 U.S.C. §1734 solely to indicate this fact. leave from the University of Texas, Austin, TX 78712. 7051 Downloaded by guest on September 27, 2021 7052 Physics: Ne'eman Proc. Natl. Acad. Sci. USA 80 (1983) not some difference in normalization, for instance, have crept picture-quantum mechanics as implying that "God plays dice"- in along the trajectories of the y quanta, so that +1 at A would or better, roulette. Should we then visualize the state vector |A) not cancel -1 at B? Are we certain that the z direction at A is flipping randomly between the states of DA, the parallelism indeed parallel to the z at B? Is a spin-scalar at the origin still provided by the connection constrains the states JB), C), etc., a scalar at A or B? It is surprising that, with so much attention and forces them to flip in unison with IA). In other words, the given to the nonlocal effects, considerations of that type were gauge field e, and its curvature or field strength f,, ensure generally not envisaged. Yet it is precisely to answer these that our quantum number go is the same everywhere and that queries-posed in a different context-that YM gauges were all fibers flip together in the quantum roulette. This is non- invented (4, 5), following Weyl (25). It is thus not surprising separability, ensured by gauge invariance. (iii) A measurement that we should now be able to claim that the geometry gen- of IA) is thus a measurement of the entire V. (iv) In a relativistic erated by such gauges (11, 12) does contain a topological mech- frame, relativistic quantum field theory being causal,.gauge in- anism realizing EPR nonseparability. Whereas the Aharonov- variance will again ensure nonseparability and this without vi- Bohm effect, monopoles, instantons, and such result from those olating causality. discrete nonsimply connected global situations, .EPR reflects parallel transport, another fundamental characteristic of fiber Comments bundles (13-15, 17-19). Paradoxes' appear when a description is-incomplete. The woman EPR is classically realized in a fiber bundle -in Houdini's trick- reappearing alive andwhole after being sawed in two--poses a paradox only as long as we are not informed Separability is a natural assumption in a picture in which space about that opening underneath her box that enables her to re- (or space-time) is uncorrelated with the relevant EPR observ- move herself from the action of the saw. Nonrelativistic single- able, as it is in nonrelativistic quantum mechanics. However, particle quantum mechanics is an approximation; it is incom- in a fiber bundle geometry, the manifold is constrained so as plete both in being nonrelativistic and in separating out a single to preserve parallelism whatever the magnitude of the base-space particle. Einstein considered separability as a plausible ap- interval. Translation from the origin, where the composite state proximation of attenuation by distance. However, this can no carries the irreducible representation D(°)mp of G (in most ex- more be-taken at face.value when the true description of nature amples-the scalar representation) has to be generated by the involves manifolds such as fiber bundles, in which space-time covariant derivative^D,4 = a,. + w',AA. Here a, is the space (or is only the base-manifold. Various schools of thought exist with respect to the incom- space-time) a , w,, represents the horizontal components of pleteness of quantum mechanics. Some claim that what is miss- the.Lie.algebra valued connection c, and A represents a basis ing is a representation of the observer's mind processes; others .of g. We work-in the product basis of Eq. 1, so that DA can act require the adjunction of an infinite replication of alternate both on the composite state and on the constituents. worlds and such. What I suggest here is that-to the extent that c EPR is concerned-the paradox arises only because of the na- The manner in which the connection at any point p(x,y) ture of the single-particle approximation (separability), an ide- of a principal bundle P(M,G,ir,.) maps the tangent manifold P* alization that is contradicted by the geometric structure of the onto the abstract (or global) Lie algebra g of G has been pre- (gauge theory) complete physics. sented elsewhere (26). x is a coordinate on M, y is a coordinate This geometric resolution of the EPR paradox fits observ- on the fiber G. In a field theory, g is given by the Noether theo- ables protected by a gauge principle. This is true of momenta rem asa space integral over P divided by the volume of G. The and of angular momenta, where gravity plays that role. It is also imap-provided. by~o is almost an inverse map to the one pro- true of the exactly conserved quantum numbers of color-SU(3) vided by the-dot (-), representing G acting on the right on P. in quantum chromodynamics and of the (spontaneously broken) P*G P degrees of freedom of quantum asthenodynamics (the unified electro-weak interactions) with the SU(2) x U(1) of Glashow, through a mapping t from g- onto P*. Almost, because the di- Weinberg, and Salam. Indeed, for quantum chromodynamics, mensionalities of P* and G are different it would be easy to imagine a gedanken triple-EPR experiment, [P*] = [M] + [G]. [2] based on deep inelastic e N scattering processes inside a nu- cleon N, with apparent causally unrelated but quantum-cor- ir is a projection (defining the vertical direction) from P onto related measurements of the color-SU(3) of the three quarks M (constrained by the color scalar property of the nucleon). The answer is not clear for a variable such as isospin or fla- V pEP,Va,a' EG -ir(pa)= ir(p) [3] vor-SU(3) (unitary spin). At the phenomenological level, these (pea) a' = pe(aa'). [4] are also gauge theories, with p, o, K*, and 4 as gauge fields. However, there is an-explicit nongauge-invariant mass term for Note that t realizes the Lie bracket (LB) operation on P* as a these connections. Parallelism is thus not guaranteed, and the Poisson bracket (PB) EPR experiment itself is thus somewhat not well-defined. V L, L' E g, t([L, L']LB) = [t(L), t(L')]PB. [5] In conclusion, EPR is realized naturally, at the classical level in a gauge theory. Nonrelativistic quantum mechanics is still an Returning to the EPR experiment, we observe that an as- incomplete theory, not in Einstein's sense but-in the sense of X sociated vector bundle V(PM,G,DA DB X ..., ir,@) realizes assuming an infinite velocity of light and a vanishing Newton's the following features naturally: (i) G acts on V equivariantly constant. Presumably, when the theory becomes relativistic and (26) so that both the composite state in D(°)mP (0) at the origin includes gravity, EPR is satisfied for most observables "natu- x = 0 and the separating constituents DA(X), DB(x'), etc., trans- rally"-i.e., we may follow then the manner in which Einstein's form parallelly and together under the action of G. The con- separability assumption is violated. straint (Eq. 1) is thus maintained whatever the magnitude of Local gauge invariance implies local unobservability of a vari-

the intervals AB, etc .... (ii) Antiparaphrasing Einstein, we able. A measurement or an alteration at x is locally unobserv- Downloaded by guest on September 27, 2021 Physics: Ne'eman Proc. Natl. Acad. Sci. USA 80 (1983) 7053 able, but the constraints on the geometric structure of the fiber 4. Yang, C. N. & Mills, R. (1954) Phys. Rev. 95, 631. bundle do transmit the relevant information over the entire base 5. Yang, C. N. & Mills, R. (1954) Phys. Rev. 96, 191-195. space. This is represented by the curvature (or the gauge field). 6. Polyakov, A. M. (1975) Phys. Lett. B 59, 82-84. 7. Belavin, A. A., Polyakov, A. M., Schwartz, A. S. & Tyupkin, Yu. At point B, the information is acknowledged, thus reproducing S. (1975) Phys. Lett. B 59j 85-87. EPR classically. All points of the base manifold thus "know" of 8. Einstein, A., Podolsky, B. & Rosen, N. (1935) Phys. Rev. 47, 777- the alteration at x, even though it cannot be known at x itself. 780. Aharonov has conjectured that, in quantum mechanics, it is the 9. Chambers, R. G. (1960) Phys. Rev. Lett. 5, 3-5. uncertainty principle that represents an analogous local partial 10. Yang, C. N. (1974) Phys. Rev. Lett. 33, 445-447. unobservability of A at x, with (only) B at x' being automatically 11. Lubkin, E. (1963) Ann. Phys. (N. Y.) 23, 233-283. 12. Wu, T. T. & Yang, C. N. (1975) Phys. Rev. D 12, 3845-3857. affected by. that measurement or alteration at A. Possibly, an 13. Kobayashi, S. & Nomizu, K. (1963) Foundations of Differential n-constituents quantum EPR could be represented by a dis- Geometry (Wiley Interscience, New York), Vol. 1. crete or lattice analog of our classical bundle. 14. Sternberg, S. (1964) Lectures on Differential Geometry (Pren- In this article, I have not touched on the other puzzling fea- tice-Hall, Englewood Cliffs, NJ). ture of quantum mechanics-i.e., the "collapse of the state- 15. Choquet-Bruhat, Y., de Witt-Morette, C. & Dillard-Bleick, M. vector." I conjecture that the nonquantum features of the ap- (1977) Analysis Manifolds and Physics (North-Holland, Amster- dam). paratus can be represented by a phase-transition description. 16. Yang, C. N. (1977) Ann. N.Y. Acad. Sci. 294, 86-97. In the gravitationally induced transformation of a "normal" 17. Drechsler, W. & Mayer, M. E. (1977) Differential Geometry and (classically describable) star into awhite dwarf or a neutron star- Gauge Theories, Springer-Verlag Lecture Notes on Physics (Spring- states in which the collapse is stopped by quantum degener- er, New York), Vol. 67. acy-we observe the transition from classical to quantum as a 18. Atiyah, M. F. (1979) Geometry of Yang-Mills Fields (Acad. Naz. critical phenomenon. The same happens in the cooling down dei Lincei Pisa). 19. Ne'eman, Y. (1979) SymetriesJauges et Varietes de Groupe (Univ. of a superfluid or a superconductor. It is thus plausible that an of Montreal Press, Montreal). appropriate generalization might work for macroscopic bodies 20. Bohm, D. & Aharonov, Y. (1957) Phys. Rev. 108, 1070-1077. with the characteristics of the Copenhagen school apparatus. 21. Bell, J. S. (1966) Rev. Mod. Phys. 38, 447-452. 22. Clauser, J. F., Home, M. A., Shimony, A. & Holt, R. A. (1969) I would like to thank F. Dyson, S. Nussinov, and Y. Aharonov for Phys. Rev. Lett. 23, 880-884. several useful discussions. I am grateful to Drs. H. Woolf, J. Bahcall, 23. Clauser, J. F. & Shimony, A. (1978) Rep. Prog. Phys. 41, 1881- and R. Dashen for-the hospitality of the Institute for Advanced Study, 1927. where this paper was conceived. 24. Aspect, A., Dalibard, J. & Roger, G. (1982) Phys. Rev. Lett. 49, 1804-1807. 1. Aharonov, Y. & Bohm, D. (1959) Phys. Rev. 115, 485-491. 25. Weyl, H. (1929) Z. Phys. 56, 330-352. 2. Dirac, P. A. M. (1931) Proc. R. Soc. London Ser. A 133, 60-72. 26. Ne'eman, Y. & Thierry-Mieg, J. (1980) Proc. Natl. Acad. Sci. USA 3. 't Hooft, G. (1974) Nucl. Phys. B 79, 276-284. 77, 720-723. Downloaded by guest on September 27, 2021