View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Directions in General Relativity, edited by B. L. Hu, M. P.Ryan and C. V. Vishveshwara (Cambridge Univ. Press, 1993), Vol. 1, Pap ers in honor of Charles Misner. REMARKS CONCERNING THE GEOMETRIES OF GRAVITY AND GAUGE FIELDS JEEVA ANANDAN DepartmentofPhysics and Astronomy, University of South Carolina, Columbia SC 29208, USA Abstract An important limitation is shown in the analogy between the Aharonov-Bohm e ect and the paral lel transport on a cone. It il lustrates a basic di erencebetween gravity and gauge elds due to the existence of the solder form for the space-time geometry. This di erence is further shown by the observability of the gravitational phase for open paths. This reinforcesaprevious suggestion that the fundamental variables for quantizing the gravitational eld are the solder form and the connection, and not the metric. 1. INTRODUCTION I recall with great pleasure the discussions which I had with Charles Misner on funda- mental asp ects of physics, such as the geometry of gravity, gauge elds, and quantum theory. In particular, I rememb er the encouragementhegavetomy somewhat un- processed by the SLAC/DESY Libraries on 29 Mar 1995. 〉 ortho dox attempts to understand the similarities and di erences b etween gauge elds and gravity from their e ects on quantum interference, and their implications to phys- ical geometry. It therefore seems appropriate to present here for his Festschrift some PostScript observations which came out of this investigation. Geometry is a part of mathematics which can b e visualized, and is intimately related to symmetries. This may explain the tremendous usefulness of geometry in physics. In section 2, I shall make some basic remarks ab out the similarities and di erences between the geometries of gravity and gauge eld. Then I shall illustrate, in section 3, an imp ortant di erence b etween them that arises due to the existence of the solder form for gravity, using the Aharonov-Bohm (AB) e ect and parallel transp ort on a cone. In section 4, I shall further illustrate this di erence by the fact that the gravitational phase for a spinless particle is observable for an op en path, unlike the GR-QC-9503055 AB e ect. This implies that the translational gauge symmetry of the gravitational eld is broken by the existence of the solder form. It is then argued that the solder form and the connection are the prop er variables for quantizing the gravitational eld. 2. LOCALITY OF GRAVITY AND GAUGE FIELDS Something which Misner emphasized to me during a conversation was the fundamen- tal role assigned to lo calityby the theory of relativity. Already in sp ecial relativity lo cality is incorp orated in the fact that signals cannot travel faster than the sp eed of light. But in general relativity, lo cality plays an even more fundamental role: The principle of equivalence states that the laws of physics are lo cally Minkowskian. Also, b ecause space-time is curved, there is no distant parallelism and vectors at two dif- ferent p oints can only b e compared by parallel transp orting them to a common p oint with resp ect to the gravitational connection. 1 These three asp ects of lo cality are also present in gauge elds which are now b eing used to describ e the three remaining fundamental interactions in physics. The prin- ciple of equivalence for gauge elds may b e stated as follows: Given any p oint p in space-time, a gauge can b e chosen so that the corresp onding connection co ecients or vector p otential vanishes at p for all elds interacting with the given gauge eld. Also, there is no distant parallelism for vectors parallel transp orted using the gauge eld connection if the curvature or the Yang-Mills eld strength is non vanishing. Contrary to what is sometimes said, gravity do es not di er fundamentally from gauge elds simply b ecause it is asso ciated with a metric. Because if the gauge group is unitary then it leaves invariant a metric in the vector space at each space-time p oint that consists of all p ossible values of any matter eld interacting with the gauge eld at that p oint. The essential di erence is that the gravitational metric can b e used to measure distances along any curve in space-time, unlike the gauge eld metric. But I shall shownow, by means of physical arguments, that this fundmental di erence between gravity and gauge elds exists even prior to intro ducing the metric. 3. AHARONOV-BOHM EFFECT AND PARALLEL TRANSPORTON A CONE It is an interesting fact that the phase shifts in quantum interference due to gravity and gauge elds are obtained in a simple manner from the distance due to the gravita- tional metric and parallel transp ort due to gravitational and gauge eld connections 2 along the interfering b eams . Conversely, the phase shifts in quantum interference 3 can b e used to de ne gauge elds and gravity . This is most easily shown for the 4 simplest gauge eld, namely the electromagnetic eld, by means of the AB e ect . We recall that the magnetic AB e ect is the phase shift in the interference of two coherent electron b eams which enclose a cylinder containing a magnetic ux. In the interference region, the wave function may b e written as (r;t)+ (r;t), where 1 2 1 and are the wave functions corresp onding to the two b eams. The intro duction of 2 the magnetic eld inside the cylinder mo di es this wave function to (r;t)= (r;t)+ F (r;t); 1 2 in an appropriate gauge, where I ie A dx ): (1) F = exp( hc Here the integral is along the curve going around the cylinder, A is the electromag- netic 4-vector p otential and e is the charge of the electron. Therefore the intensity 2 distribution j (r;t)j in the interference region is mo di ed in an apparantly non lo cal wayby the magnetic ux via F even though the magnetic and electric eld strengths vanish everywhere along the b eams. But this phenomenon is not surprising when we realize the analogy with the geometry 5 of a cone . The cone may b e formed by taking a at sheet of pap er b ounded bytwo straight lines making an angle and identifying the two straight lines (Fig. 1a); we denote this cone by C .For 0 2 , this is what we do when we make a cone by rolling this at sheet so that these two lines coincide to form one of the generators of the cone. Since the pap er is not stretched or compressed during this pro cess, a cone has no intrinsic curvature except at the ap ex, which can b e smo othed out so that the curvature is nite there. In the multiply connected geometry around the ap ex, the intrinsic curvature is zero everywhere, same as the at geometry of the sheet which was rolled up to b e the cone. In particular, a vector is parallel transp orted likeon the at sheet. But a vector V parallel transp orted around a closed curve drawn on the curvature free region of the cone so as to enclose the ap ex undergo es a rotation by the angle , which is the holonomy transformation asso ciated with this curve. If the curvature at the ap ex is regarded as analogous to the magnetic eld in the cylinder then the zero intrinsic curvature everywhere else corresp onds to the vanishing of magnetic eld strength outside the cylinder. Then V moving in a curvature free region is analogous to b eams traveling in a eld free region. The rotation by the 0 angle which relates V and V (Fig. 1a) is analogous to the phase di erence b etween the two b eams due to F . This suggests that the electromagnetic eld maybea connection for parallel transp orting the value of the wave function and the AB e ect arises b ecause a wave function when parallel transp orted around the closed curve , gets multiplied by F . The electric and magnetic eld strengths at each space-time Figure 1 a) Analogy between the Aharonov-Bohm e ect and paral lel transport on a cone. The 0 cone may be obtained by identifying the lines OA and OA on a at sheet. There- fore, the vector V paral lel transportedfrom B around the cone would come back to B 0 0 (identi ed with B )asV rotated by the angle . This is analogous to the AB phase shift with the magnetic eld corresponding to the curvature at the apex of the cone. b) The limitation of this analogy when is changedto+2 by adding an extra sheet 0 of paper. The vector paral lel transported along the closed curve BCDEB rotates by +2 with respect to the tangent vector to the curve. This enables one to distinguish this cone from the earlier one. This is unlike the AB e ect which cannot distinguish between two enclosed magnetic uxes that di er by one quantum of ux. p oint then constitutes the curvature of this connection at this p oint. Thus the phase factor (1) is the holonomy transformation asso ciated with for this connection. The statement that the electromagnetic eld is a gauge eld is the same as saying that it is a connection as describ ed ab ove.