Quantum mechanics as a spontaneously broken gauge theory on a U(1) gerbe
QQUANTUM MECHANICS AS A SPONTANEOUSLYBROKEN GAUGE THEORY ON A U(1) GERBE
Jos´e M. Isidro
Grupo de Modelizaci´on Interdisciplinar, Instituto de Matem´atica Pura y Aplicada,Universidad Polit´ecnica de Valencia, Valencia 46022, SpainMax–Planck–Institut f¨ur Gravitationsphysik, Albert–Einstein–Institut,D–14476 Golm, Germany [email protected]
October 23, 2018
Abstract
Any quantum–mechanical system possesses a U(1) gerbe naturally definedon configuration space. Acting on Feynman’s kernel exp(i S/ (cid:126) ) , this U(1) symmetryallows one to arbitrarily pick the origin for the classical action S , on a point–by–pointbasis on configuration space. This is equivalent to the statement that quantum me-chanics is a U(1) gauge theory. Unlike Yang–Mills theories, however, the geometry ofthis gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since thisgauge symmetry is spontaneously broken, an analogue of the Higgs mechanism mustbe present. We prove that a Heisenberg–like noncommutativity for the space coordi-nates is responsible for the breaking. This allows to interpret the noncommutativity ofspace coordinates as a Higgs mechanism on the quantum–mechanical U(1) gerbe. Contents
A Appendix: computing the trivialisation 15
A.1 The constant potential . . . . . . . . . . . . . . . . . . . . . . . . . . 151 a r X i v : . [ m a t h - ph ] A ug .2 The linear potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A.3 The quadratic potential . . . . . . . . . . . . . . . . . . . . . . . . . 17 Let M be an n –dimensional spacetime manifold endowed with the the metric tensor g µν . Let x µ , µ = 1 , . . . , n , be local coordinates on M . The possibility of measuringthe infinitesimal distance d s = g µν d x µ d x ν (1)between two points on M rests on the assumption that the corresponding coordinatescan be simultaneously measured with infinite accuracy, so one can have ∆ x µ = 0 (2)simultaneously for all µ = 1 , . . . , n . In quantum–mechanical language one wouldrecast this assumption as [ˆ x µ , ˆ x ν ] = ˆ x µ ˆ x ν − ˆ x ν ˆ x µ = 0 , (3)where ˆ x µ is a quantum operator whose classical limit is the coordinate function x µ .The vanishing of the above commutator expresses two alternative, though essentiallyequivalent, statements, one of physical content, the other geometrical. Physically itexpresses the absence of magnetic fields across the µ, ν directions [1]. Geometricallyit expresses the fact that the multiplication law on the algebra of functions on the space M is commutative.All modern theories of quantum gravity [2] share the common feature that a mini-mal length scale, the Planck length L P , exists on spacetime, ∆ x µ ≥ L P , (4)so L P effectively becomes the shortest possible distance, and its square L P becomesproportional to the quantum of area. This coarse graining of a spacetime continuum M can be mimicked, in noncommutative geometry [3], by noncommuting operatorcoordinates ˆ x µ acting as Hermitean operators on Hilbert space H . The ˆ x µ satisfy [ˆ x µ , ˆ x ν ] = i aθ µν , (5)with θ µν a constant, real, dimensionless antisymmetric tensor. Here a > is a funda-mental area scale, such that lim a → [ˆ x µ , ˆ x ν ] = 0 . (6)Moreover, in the limit a → , one can identify (possibly up to some singular renormali-sation factor Z ) the operator ˆ x µ on H with the function x µ on M . Since the Heisenberguncertainty relations corresponding to (5) imply ∆ˆ x µ ∆ˆ x ν ≥ a | θ µν | , (7)2he above statement concerning the coarse graining of M follows. Up to possible nu-merical factors C one can therefore set a = CL P . (8)It has been argued [4] that the existence of a fundamental length scale L P on M implies modifying the spacetime metric according to the rule d s −→ d s + L P , (9)so L P effectively becomes the shortest possible distance. One can also prove [4] thatmodifying the spacetime interval according to (9) is equivalent to requiring invarianceof a field theory under the following exchange of short and long distances: d s ←→ L P d s . (10)Further consequences of the exchange (10) have been reported in ref. [5].On the other hand, we have in ref. [6] shown that the existence of a minimal lengthscale L P is equivalent to the exchange S (cid:126) ←→ (cid:126) S (11)in Feynman’s exponential of the action integral S : exp (cid:18) i S (cid:126) (cid:19) ←→ exp (cid:18) i (cid:126) S (cid:19) . (12)In other words, the duality (10) is equivalent to the duality (12). Since the equationsof motion that follow from the variation of S/ (cid:126) are the same as those derived fromthe variation of (cid:126) /S , classically there is no difference between S/ (cid:126) and (cid:126) /S . We willrefer to the exchange (11) as semiclassical vs. strong–quantum duality . This simple Z –transformation has been extended [6] to larger duality groups G such as SL (2 , Z ) , SL (2 , R ) and SL (2 , C ) . Examples of the semiclassical vs. strong–quantum duality(11) have appeared under different, though essentially equivalent, guises, in refs. [7];see [8] for related works.Examining the relation between the noncommutativity (5) of the space coordinatesand the quantum of area (8) one realises that eqns. (5) and (8) are in fact equivalent.The commutation relations (5) imply the existence of a quantum of area: by (5) onehas ∆ˆ x j ∼ L P , hence a quantum of area must exist and be proportional to L P . Con-versely, let a quantum of area ∆ˆ x j ∆ˆ x k ∼ L P be given. The latter could not exist ona spacetime continuum whose coordinates all commute, since then we would alwayshave ∆ˆ x j ∆ˆ x k = 0 . The simplest noncommutativity giving rise to a quantum of areais (5); more general types of noncommutativity can also be considered [9]. The aboveequivalence between eqns. (5) and (8) is intuitively obvious, but it will be very in-structive to recast it in the geometrical language of gerbes [10]. We have in ref. [11]succeeded in interpreting quantum mechanics as a U(1) gauge theory on phase space.3owever, unlike Yang–Mills theories, the gauge symmetry is not expressed geometri-cally by means of a connection 1–form and its corresponding curvature 2–form on afibre bundle. Rather, the appropriate geometrical setup will be provided by a gerbe.The first goal of this paper is to break the U(1) symmetry on the quantum–mechanicalgerbe constructed in ref. [11]. The breaking will occur via an analogue of the usualHiggs mechanism of Yang–Mills theory, as adapted now to the fact that gerbes liveone step up from bundles (for the geometrical aspects of the Higgs mechanism see ref.[12]). This breaking is necessary since the exchange (10), or its equivalent (12) onthe U(1) gerbe, is certainly not realised in Nature as observed at low energies. How-ever dualities such as (10) and (12) are to be expected [4] within the realm of quantumgravity. Moreover, quantum–gravity effects have also been conjectured to be relevantat astrophysical scales [13]; gravity itself can be understood as arising from the break-ing of local Lorentz symmetry [14]. All this evidence strongly suggests a study of thesymmetry–breaking mechanism in our setup.As a second goal of this article, we will prove that a space noncommutativity ofthe type (5) provides the gerbe analogue of the Higgs mechanism in Yang–Mills theory .Our previous results of ref. [11] were deduced on phase space, where an interesting linkcould be established with the phase–space formulation of quantum mechanics [15]. Inthe present paper we will work on configuration space instead.To summarise, we will see that the requirement of semiclassical vs. strong–quantumduality (12) will lead to a quantisation of spacetime, and viceversa. Thus the gerbe ap-proach to nonrelativistic quantum mechanics analysed here, duly generalised to therelativistic case, can provide an interesting route towards a quantum theory of gravity. In ref. [11] we have given a detailed construction of a quantum–mechanical gerbeon phase space. In what follows we briefly recall its main features and adapt it toconfiguration space.
It is well known [16] that a unitary line bundle on a base manifold M is a 1–cocycle λ ∈ H ( M , C ∞ (U(1))) . The latter is the first ˇCech cohomology group of M withcoefficients in the sheaf of germs of smooth, U(1)–valued functions. Let { U α } be agood cover of M by open sets U α . Then the bundle is determined by a collection ofU(1)–valued transition functions defined on each 2–fold overlap λ α α : U α ∩ U α −→ U(1) (13)satisfying λ α α = λ − α α , (14)as well as the 1–cocycle condition λ α α λ α α λ α α = 1 on U α ∩ U α ∩ U α . (15)4 gerbe is defined as a 2–cocycle g ∈ H ( M , C ∞ (U(1))) . This means that wehave a collection { g α α α } of maps defined on each 3–fold overlap on M g α α α : U α ∩ U α ∩ U α −→ U(1) (16)satisfying g α α α = g − α α α = g − α α α = g − α α α , (17)as well as the 2–cocycle condition g α α α g − α α α g α α α g − α α α = 1 on U α ∩ U α ∩ U α ∩ U α . (18)Now g is a 2–coboundary in ˇCech cohomology whenever it holds that g α α α = τ α α τ α α τ α α (19)for a certain collection { τ α α } of U(1)–valued functions τ α α on U α ∩ U α suchthat τ α α = τ − α α . The collection { τ α α } is called a trivialisation of the gerbe. Onecan prove that over any given open set U α of the cover { U α } there always exists atrivialisation of the gerbe.On a gerbe specified by the 2–cocycle g α α α , a connection is specified by forms A, B, H satisfying H | U α = d B α (20) B α − B α = d A α α (21) A α α + A α α + A α α = g − α α α d g α α α . (22)The 3–form H is the curvature of the gerbe connection. The latter is called flat if H = 0 . Let an action integral S be given for a point particle on the spacetime M . Let us furtherassume that the latter factorises, at least locally, as the product of the time axis R anda configuration space F . Coordinates x µ ( α ) on the local chart labelled by α thereforedecompose as ( t α , q jα ) , with j = 1 , . . . , n − . This latter index will be suppressed inwhat follows. Let any two points q α , q α be given on F , with local charts U α , U α centred around them. Charts for the time coordinate will not be indicated explicitlyunless necessary. Moreover, let L α α be an oriented path connecting q α to q α astime runs from t α to t α . We define a α α as the following functional integral over allsuch trajectories L α α : a α α ∼ (cid:90) D L α α exp (cid:20) i (cid:126) S ( L α α ) (cid:21) . (23)Throughout this paper, the ∼ sign will stand for proportionality : path integrals aredefined up to some (usually divergent) normalisation. However all such normalisationfactors will cancel in the ratios of path integrals that we are interested in. The argument5f the exponential in eqn. (23) contains the action S evaluated along the path L α α .Thus a α α is proportional to the probability amplitude for the particle to start at q α and finish at q α , i.e. , it is proportional to the propagator G ( q α , t α ; q α , t α ) : (cid:90) D L α α exp (cid:20) i (cid:126) S ( L α α ) (cid:21) ∼ G ( q α , t α ; q α , t α ) . (24)Now assume that U α ∩ U α is nonempty, U α α := U α ∩ U α (cid:54) = φ. (25)and define, for ( q α , q α , q α ) ∈ U α α × U α × U α , τ (cid:48) α α : U α α × U α × U α −→ C τ (cid:48) α α := a α α a α α . (26)Thus τ (cid:48) α α is proportional to the probability amplitude for the following transition:starting at q α , the particle reaches q α after traversing the variable midpoint q α . Wehave τ (cid:48) α α ∼ (cid:90) D L α α ( α ) exp (cid:26) i (cid:126) S [ L α α ( α )] (cid:27) (27) = (cid:90) D L α α exp (cid:20) i (cid:126) S ( L α α ) (cid:21) (cid:90) D L α α exp (cid:20) i (cid:126) S ( L α α ) (cid:21) ∼ G ( q α , t α ; q α , t α ) G ( q α , t α ; q α , t α ) . As it stands, τ (cid:48) α α is a function on U α α × U α × U α because of its dependence onthe endpoints q α and q α , which are being kept fixed. A true trivialisation should be afunction on the double overlap U α α only. However we can integrate τ (cid:48) α α over q α and q α in order to eliminate this dependence. We thus define ˜ τ α α : U α α −→ C , (28) ˜ τ α α := (cid:90) d q α d q α τ (cid:48) α α = (cid:90) d q α d q α G ( q α , t α ; q α , t α ) G ( q α , t α ; q α , t α ) . Since a trivialisation must be a U(1)–valued function, we finally define τ α α : U α α −→ U(1) , τ α α := ˜ τ α α | ˜ τ α α | , (29)whenever ˜ τ α α is nonvanishing. One can verify that τ α α qualifies as a trivialisa-tion on F . Physically, this trivialisation is interpreted as the U(1)–valued phase of theprobability amplitude for the particle to start at any initial point in the chart U α andto reach any final point in the chart U α , while traversing the midpoint q α ∈ U α α .Observe that (28) contains a Riemann volume integral while (27) contains a Feynmanpath integral. Notice also that (29) depends parametrically on the times t α , t α and6 α ; it will also depend parametrically on whatever other parameters the action S maycontain such as masses, forces, frequencies, coupling constants, etc . However, as thetrivialisation of a gerbe over F , τ α α depends only on the point q α ∈ U α α as itshould.The trivialisations corresponding to a number of cases are worked out expliciltyin the appendix. These examples prove that, at least up to (and including) quadraticterms, which is the degree of approximation we will keep throughout, whatever zeroesthe propagators may have, these zeroes will all cancel in the end. Thus the trivialisation,being a U(1)–phase, is always well defined. One can think of τ α α ( q α ) as the U(1)–phase of the averaged ( i.e. , integrated) probability amplitude for the particle to startsomewhere in U α and finish somewhere in U α while crossing q α ∈ U α α . Next consider three points and their respective charts q α ∈ U α , q α ∈ U α , q α ∈ U α (30)such that the triple overlap U α ∩ U α ∩ U α is nonempty, U α α α := U α ∩ U α ∩ U α (cid:54) = φ. (31)Once the trivialisation (28) is known, the 2–cocycle g α α α defining a gerbe on F isgiven by (19): g α α α : U α α α −→ U(1) g α α α ( q α ) := τ α α ( q α ) τ α α ( q α ) τ α α ( q α ) , (32)where all three τ ’s on the right–hand side are, by definition, evaluated at the samevariable midpoint q α ∈ U α α α . (33)Being U(1)–valued, the 2–cocycle (32) can be expressed as the quotient of a complexfunction ˜ g by its modulus, g α α α ( q α ) = ˜ g α α α ( q α ) | ˜ g α α α ( q α ) | . (34)By eqns. (28) and (32) we have ˜ g α α α ( q α ) (35) ∼ (cid:90) d q α d q α G ( q α , t α ; q α , t α ) G ( q α , t α ; q α , t α ) × (cid:90) d q (cid:48) α d q α G ( q (cid:48) α , t (cid:48) α ; q α , t (cid:48) α ) G ( q α , t (cid:48) α ; q α , t α ) × (cid:90) d q (cid:48) α d q (cid:48) α G ( q (cid:48) α , t (cid:48) α ; q α , t (cid:48)(cid:48) α ) G ( q α , t (cid:48)(cid:48) α ; q (cid:48) α , t (cid:48) α ) , t α < t α < t α < t (cid:48) α < t (cid:48) α < t α < t (cid:48) α < t (cid:48)(cid:48) α < t (cid:48) α . (36)Thus g α α α ( q α ) equals the U(1)–phase of the probability amplitude for the fol-lowing transition: starting anywhere in U α (say, at q α ), the particle crosses q α onits way to some q α ∈ U α ; the points q α and q α are integrated over. Next, start-ing at some q (cid:48) α ∈ U α , the particle crosses the same q α again on its way to some q α ∈ U α ; the points q (cid:48) α and q α are also integrated over. Finally, from q (cid:48) α ∈ U α ittraverses q α once more before finally reaching some q (cid:48) α ∈ U α ; the points q (cid:48) α and q (cid:48) α are also integrated over.It must be observed that the points q (cid:48) α , q (cid:48) α and q (cid:48) α are not necessarily identicalwith q α , q α and q α , respectively. Thus the transition considered does not necessarilydefine a closed path on F , although all such paths traverse q α . Moreover, condition(36) implies that the complete trajectory is never closed as a path on F × R . However,in the particular case that one or more of the equalities q α = q (cid:48) α , q α = q (cid:48) α and q α = q (cid:48) α does not hold, we can always connect the points q α j and q (cid:48) α j within thecorresponding U α j , so as to complete a closed loop on F . This closed loop is theprojection, onto F , of an open loop on F × R . It is possible to complete such anopen path to a closed loop because the transition amplitides considered above are allintegrated over the endpoints q α j and q (cid:48) α j . In so doing we obtain a closed loop on F such as that in the figure: L α α α ( α ) := L α α ( α ) + L α α ( α ) + L α α ( α ) . (37)Recalling that the propagator can be expressed as the functional integral (24), we con-clude that (35) can be expressed as a functional integral over all closed loops on F ofthe type (37): ˜ g α α α ( q α ) ∼ (cid:90) D L α α α ( α ) exp (cid:26) i (cid:126) S [ L α α α ( α )] (cid:27) . (38)From now on we will restrict our attention to closed loops on F of the type (37). Next we will recast eqn. (38) into an equivalent, but more useful, expression. Givena closed loop L , let S ⊂ F be a 2–dimensional surface with boundary such that ∂ S = L .By Stokes’ theorem, S ( L ) = (cid:90) L L d t = (cid:90) ∂ S L d t = (cid:90) S d L ∧ d t. (39)Any surface S such that ∂ S = L will satisfy eqn. (39) because the integrand d L ∧ d t is closed. Let us now choose S to bound a closed loop L α α α ( α ) as in eqn.(37). Consider the first half of the leg L α α ( α ) , denoted L α α ( α ) . The latterruns from α to α . Consider also the second half of the leg L α α ( α ) , denoted The above discussion also settles an apparent discrepancy between the definition of the trivialisationgiven here and that given in ref. [11]. The correct definition of the trivialisation is the one given in section2.2 here. However the 2–cocycle (34), (38) obtained from the trivialisation of section 2.2, and therefore thegerbe itself, coincides with that of ref. [11]. (cid:48) L α α ( α ) , with a prime to remind us that it is the second half: it runs back from α to α . The sum of these two half legs, L α α ( α ) + 12 (cid:48) L α α ( α ) , (40)completes one roundtrip and it will, as a rule, enclose an area S α ( α ) , unless thepath from α to α happens to coincide exactly with the path from α to α : ∂ S α ( α ) = 12 L α α ( α ) + 12 (cid:48) L α α ( α ) . (41)Analogous conclusions apply to the other half legs (cid:48) L α α ( α ) , L α α ( α ) , L α α ( α ) and (cid:48) L α α ( α ) under cyclic permutations of 1,2,3 in the ˇCech in-dices α , α and α : ∂ S α ( α ) = 12 L α α ( α ) + 12 (cid:48) L α α ( α ) , (42) ∂ S α ( α ) = 12 L α α ( α ) + 12 (cid:48) L α α ( α ) . (43)The boundaries of the three surfaces S α ( α ) , S α ( α ) and S α ( α ) all passthrough the variable midpoint α , although we will no longer indicate this explicitly.We define their connected sum S α α α := S α + S α + S α . (44)In this way we have L α α α = ∂ S α α α = ∂ S α + ∂ S α + ∂ S α . (45)It must be borne in mind that L α α α is a function of the variable midpoint α ∈ U α α α , even if we no longer indicate this explicitly. Eventually one, two or perhapsall three of S α , S α and S α may degenerate to a curve connecting the midpoint α with α , α or α , respectively. Whenever such is the case for all three surfaces, theclosed trajectory L α α α cannot be expressed as the boundary of a 2–dimensionalsurface S α α α . In what follows we will however exclude this latter possibility, sothat at least one of the three surfaces on the right–hand side of (44) does not degenerateto a curve.In general we will not be able to compute the functional integral (38) exactly. How-ever we can gain some insight from a steepest–descent approximation [17], the detailsof which have been worked out in ref. [11]. We find g (0) α α α ( q α ) = exp (cid:26) i (cid:126) S (cid:104) L (0) α α α ( α ) (cid:105)(cid:27) , (46)the superindex (0) standing for evaluation at the extremal . The latter is that path which,meeting the requirements stated after eqn. (37), minimises the action S . To summarise,9y eqns. (38), (39), (44), (45) and (46), we can write the steepest–descent approxima-tion to the 2–cocycle as g (0) α α α = exp (cid:32) i (cid:126) (cid:90) S (0) α α α d L ∧ d t (cid:33) , (47)where S (0) α α α is a minimal surface for the integrand d L ∧ d t . We will henceforthdrop the superindex (0) , with the understanding that all our computations have beenperformed in the steepest–descent approximation. We can use eqns. (46) and (47) in order to compute the connection, at least to the sameorder of accuracy as the 2–cocycle itself. We find A α α = i (cid:126) ( L d t ) α α , (48) B α − B α = d A α α = i (cid:126) (d L ∧ d t ) α α , (49) H | U α = d B α . (50)A comment is in order. The potential A is supposed to be a 1–form on configurationspace F , on which the gerbe is defined. As it stands in (48), due to the factor d t , A is a1–form on F × R . If ι : F → F × R denotes the natural inclusion, the 1–form A in (48)is to be understood as its pullback ι ∗ ( L d t ) onto F . We will however continue to writeit as L d t . Let us perform the transformation L d t −→ L d t + d f, f ∈ C ∞ ( F ) , (51)where f is an arbitrary function on F with the dimensions of an action. The abovetransformation does not alter the dynamics defined by S : it amounts to shifting S by aconstant C , S −→ S + C, C := (cid:90) d f. (52)The way the transformation (51) acts on the quantum theory is well known. In theWKB approximation, the wavefunction reads [15] ψ WKB = R exp (cid:18) i (cid:126) S (cid:19) (53)10or some amplitude R . Thus the transformation (51) multiplies the WKB wavefunc-tion ψ WKB and, more generally, any wavefunction ψ , by the constant phase factor exp (i C/ (cid:126) ) : ψ −→ exp (cid:18) i (cid:126) C (cid:19) ψ. (54)Gauging the rigid symmetry (54) one obtains the transformation law ψ −→ exp (cid:18) i (cid:126) f (cid:19) ψ, f ∈ C ∞ ( F ) , (55) f being an arbitrary function on configuration space, with the dimensions of an action.In the case of a gerbe over phase space, as in ref. [11], the U(1) symmetry on the gerbeimplies the possibility of performing the local gauge transformations (55). Analogousconclusions continue to hold in our case, where the gerbe is defined over configurationspace F ; see ref. [11] for further details. In particular, for the transformation (55) to bean invariance of the theory, all derivatives within the action S are to be covariantisedby means of the connection 1–form A of eqn. (48). The U(1) symmetry (55) on the gerbe is spontaneously broken. If this symmetry wereunbroken, then in particular the duality (12) between the semiclassical and the strongquantum regimes, or its equivalent (10) between long and short distances, would bemanifest. This is certainly not the case in Nature as observed at low energies, al-though it has been suggested [4] that effects such as the dualities (10) and (12) areto be expected within quantum gravity. The breaking occurs via a mechanism thatis analogous to the Higgs mechanism of Yang–Mills theory. However, since gerbesfall into a category that is one step up from that of fibre bundles, the details of thesymmetry–breaking mechanism are different here. For the breaking to take place, acertain field must develop a vacuum expectation value equal to Planck’s constant. Thisis so because quantisation is due to a nonvanishing value for (cid:126) , and we are interpretingquantum mechanics as a gauge theory. Moreover, whatever nonvanishing value (cid:126) maytake on, different numerical values for Planck’s constant lead to different quantum the-ories. A specific choice of one particular value for (cid:126) picks one, and only one, quantumtheory out of the many that are possible before the U(1) symmetry is broken.Consider the connection 1–form ˆ A on the gerbe. As usual, the caret reminds usthat ˆ A is a quantum operator corresponding to the classical field A . By eqn. (48), ˆ A isproportional to the operator ˆ L d t . The expectation value (cid:104) ˆ L d t (cid:105) can be obtained as theintegral over a certain path L i , the latter playing the role of a certain vacuum state: (cid:104) ˆ L d t (cid:105) L i := (cid:90) L i L d t = (cid:126) i . (56)In principle each path L i , or vacuum state, produces a different value for (cid:126) i . BecauseFeyman’s kernel is exp (i S/ (cid:126) ) , before symmetry breaking there is a whole U(1)’s worth11f equivalent vacua. Now the vacuum state L phys actually picked by Nature gives riseto the physical value (cid:126) phys of Planck’s constant as observed in our world: (cid:104) ˆ L d t (cid:105) L phys = (cid:90) L phys L d t = (cid:126) phys . (57)The corresponding L phys must have a length ∼ O ( L P ) . Our notations stress the dif-ference between (cid:126) i as a variable parameter and (cid:126) phys , the latter being the particularvalue assumed by that parameter in the actual world we live in. Eqn. (57) expressesthe breaking of the U(1) symmetry on the gerbe.We can also recast (56) and (57) in terms of surfaces S and 2–forms: (cid:104) d ˆ L ∧ d t (cid:105) S i := (cid:90) S i d L ∧ d t = (cid:126) i . (58)Again each surface S i , or vacuum state, produces a different value for (cid:126) i . Also, thevacuum state S phys actually picked by Nature must have an area ∼ O ( L P ) and be suchthat (cid:104) d ˆ L ∧ d t (cid:105) S phys = (cid:90) S phys d L ∧ d t = (cid:126) phys . (59)By eqn. (49), the above can also be expressed in terms of the Neveu–Schwarz operator2–form ˆ B . If the surfaces S i have boundaries ∂ S i = L i , then eqns. (59) and (58) arestrictly equivalent to (57) and (56), respectively. However the convenience of usingsurfaces S rather than paths L will become clear presently. We conclude that a nonva-nishing value for the quantum of action (cid:126) is equivalent to a nonvanishing quantum oflength proportional to L P , or to a nonvanishing quantum of area proportional to L P .We started off with a configuration space F whose coordinates q j were commuta-tive. Next we constructed a U(1) gerbe over F . The U(1) symmetry on the latter allowedus to arbitrarily pick, on a point–by–point basis, the zero point for the action integral S .As proved in ref. [11] and summarised in section 3.1, this symmetry rendered notionslike semiclassical approximation or strong–quantum regime meaningless. Finally weobserved that the U(1) symmetry must be spontaneously broken at low energies, wherethe above notions do have a definite meaning. A quantum of area results as a conse-quence, which can only exist on a noncommutative space. We can therefore state that the Higgs mechanism on the U(1) gerbe over configuration space F renders the latternoncommutative .On the other hand, as observed at low energies, space coordinates are definitelycommutative, while they are expected to turn noncommutative at an energy scale aroundthat of quantum gravity. The whole situation can be summarised in the diagram F ( (cid:126) ) −→ F (cid:63) ( (cid:126) = (cid:126) phys ) −→ F ( (cid:126) phys → . (60)The first arrow stands for the Higgs mechanism described above. It represents the pas-sage from the commutative configuration space F ( (cid:126) ) , where no value for (cid:126) has beenspecified yet, to the noncommutative space F (cid:63) ( (cid:126) = (cid:126) phys ) , on which a specific value (cid:126) phys for (cid:126) has been selected. The (cid:63) in the notation stresses the fact that the multi-plication law now is the noncommutative (cid:63) –product [3]. The second arrow represents12he passage to the limit (cid:126) phys → , in which the (cid:63) –product on F (cid:63) becomes the usual,pointwise, commutative multiplication law on the commutative space F . This is thepassage from the high–energy world F (cid:63) ( (cid:126) = (cid:126) phys ) , where quantum–gravity effectsare expected to be relevant, to the low–energy world F ( (cid:126) phys → we live in, wheresuch effects can be neglected. Strictly speaking, a quantum of area makes sense only on a noncommutative space;commutative continua do not allow for such a coarse graining, since infinitesimalscan be made as small as one pleases. Therefore a nonvanishing quantum of area isa consequence of the nonvanishing of the noncommutativity parameter θ ij . The un-certainty principle (7) on configuration space then follows immediately. This is whereone advantage of using surfaces rather than loops becomes apparent: by eqn. (49), thevacuum expectation value (59) can be related to the vacuum expectation value of theNeveu–Schwarz 2–form operator ˆ B . A choice of gauge ensures ˆ B α = 0 , while thecaret can be removed by integrating over the surface S phys . If we take the latter asspanning the j , k spatial dimensions, this integral is proportional to θ − jk , which is the(inverse) noncommutativity parameter.However, there is one fundamental difference between the uncertainty principle onconfiguration space and the usual uncertainty principle on phase space. Namely, thelatter is the result of rewriting the classical Poisson brackets { q, p } = 1 in terms ofquantum commutators, while the uncertainty principle on configuration space is a con-sequence of the breaking of the U(1) symmetry on the gerbe. Thus, while Heisenberg’sprinciple on phase space follows from the kinematic equation [ˆ q, ˆ p ] = i (cid:126) , the uncer-tainty principle (7) on configuration space involves (cid:126) phys as a dynamically generatedquantum scale . As such, Planck’s constant will be subject to a renormalisation–grouplaw, like the Yang–Mills coupling constant. In particular, the value of (cid:126) may dependon the scale. This is in perfect agreement with the conclusions of ref. [18] regardingPlanck’s constant, and also with those of ref. [13] regarding Newton’s constant.To finish this section we would like to comment on the Higgs mechanism on phasespace. In the limit L P → , configuration space becomes commutative, while phasespace retains a nonvanishing commutator (or Poisson brackets) between coordinatesand momenta. This is so because [ˆ q j , ˆ q k ] ∼ L P , while [ˆ q j , ˆ p k ] is order zero in L P . Next we would like to relate area quantisation to the quantised characteristic class forthe gerbe.It follows from eqn. (49) that d B α = d B α . This implies that the 3–form fieldstrength H , contrary to the 2–form potential B , is globally defined on F . Now the deRham cohomology class [ H ] of the 3–form H is quantised [10]: [ H ] ∈ H ( F , π i Z ) , (61)13 .e. , π i (cid:90) V (cid:48) H ∈ Z , ∂ V (cid:48) = 0 , (62)for all 3–dimensional volumes V (cid:48) ⊂ F such that ∂ V (cid:48) = 0 .Consider now a 3–dimensional volume V ⊂ F whose boundary is a 2–dimensionalclosed surface S . If V is connected and simply connected we may, without loss of gen-erality, take V to be a solid ball, so S = ∂ V is a sphere. Let us cover S by stereographicprojection. This gives us two coordinate charts, respectively centred around the northand south poles on the sphere. Each chart is diffeomorphic to a copy of the plane R .Each plane covers the whole sphere S with the exception of the opposite pole. Theintersection of these two charts is the whole sphere S punctured at its north and southpoles. Let us embed the chart R α centred at the north pole within the open set U α , i.e. , R α ⊂ U α , if necessary by means of some diffeomorphism. Analogously, for thesouth pole we have R α ⊂ U α . There is also no loss of generality in assuming thatonly two points on the sphere S (the north and south poles) remain outside the 2–foldoverlap U α ∩ U α . By Stokes’ theorem, (cid:90) V H = (cid:90) V d B = (cid:90) ∂ V B = (cid:90) S B = (cid:90) R α B − (cid:90) R α B, (63)and, by eqn. (49), (cid:90) V H = i (cid:126) (cid:90) R −{ } d L ∧ d t, (64)where R − { } denotes either one of our two charts, punctured at its correspondingorigin. Now R − { } falls short of covering the whole sphere S by just two points (thenorth and south poles), and the latter have zero measure. Excluding cases where theintegrand is supported on isolated points such as the poles, we may just as well write (cid:90) V H = i (cid:126) (cid:90) S d L ∧ d t, ∂ V = S . (65)Eqn. (65) is analogous to the Gauss law in electrostatics, with H replacing the electriccharge density 3–form and id L∧ d t/ (cid:126) replacing the corresponding surface flux 2–form.Now the quantisation condition (61) on [ H ] applies to closed volumes, while (65)refers to volumes bounded by a surface. However it seems reasonable to conjecturethat (61) should be related to some quantisation condition on the surface integral of id L ∧ d t/ (cid:126) . Since the surface integral of id L ∧ d t/ (cid:126) is related to that of the Neveu–Schwarz field B , the vacuum expectation value for ˆ B will be quantised: a fact that isalready known to us from the foregoing discussion. Specifically: if one postulates thequantisation condition i (cid:126) [d L ∧ d t ] ∈ H ( F , π i Z ) , (66)then the quantisation condition (61) follows, and viceversa. To prove this, consider twovolumes V , V such that ∂ V = S = − ∂ V , and such that glued together along theircommon boundary one obtains a V (cid:48) without boundary. Then π (cid:126) (cid:90) S d L ∧ d t = 12 π (cid:126) (cid:90) S d L ∧ d t + 12 π (cid:126) (cid:90) S d L ∧ d t π (cid:126) (cid:90) ∂ V d L ∧ d t − π (cid:126) (cid:90) ∂ V d L ∧ d t = 12 π i (cid:90) V H + 12 π i (cid:90) V H = 12 π i (cid:90) V (cid:48) H. (67)Now the last term above is an integer if and only if also the first term is an integer. Thisproves that (66) and (61) are equivalent: area quantisation on configuration space anda quantised characterictic class for the gerbe are equivalent statements.We can return to eqn. (47) and rewrite the 2–cocycle using (66): g α α α = exp (i πn α α α ) , n α α α ∈ Z . (68)In the particular case of the free particle we conclude, by (70) below, that n α α α mustbe even: n α α α = 2 k α α α , for some k α α α ∈ Z . A Appendix: computing the trivialisation
Let us work out the trivialisation explicitly for the case of a particle on the manifold F × R . Assume that local charts on F are diffeomorphic to R d , where d = n − . Thissimplifying assumption allows one to perform all computations explicitly. In whatfollows, our propagators are normalised as in ref. [19]. However it must be bornein mind that we are interested only in the nonconstant U(1)–valued phase of the finalresult. A.1 The constant potential
The propagator for a free particle is G ( q α , t α ; q α , t α ) = (cid:20) m π i (cid:126) ( t α − t α ) (cid:21) d/ exp (cid:20) i m (cid:126) ( q α − q α ) t α − t α (cid:21) . (69)By eqn. (28) ˜ τ α α = (cid:90) d q α d q α G ( q α , t α ; q α , t α ) G ( q α , t α ; q α , t α )= (cid:20) m π i (cid:126) ( t α − t α ) (cid:21) d/ (cid:90) d q α exp (cid:20) i m (cid:126) ( q α − q α ) t α − t α (cid:21) × (cid:20) m π i (cid:126) ( t α − t α ) (cid:21) d/ (cid:90) d q α exp (cid:20) i m (cid:126) ( q α − q α ) t α − t α (cid:21) = 1 . (70)Hence the free particle has a trivial, i.e. , constant, trivialisation.15 .2 The linear potential Consider a particle acted on by a constant force F . The propagator then reads G ( q α , t α ; q α , t α ) = (cid:20) m π i (cid:126) ( t α − t α ) (cid:21) d/ (71) × exp (cid:26) i (cid:126) (cid:20) m q α − q α ) t α − t α + F t α − t α )( q α + q α ) − F m ( t α − t α ) (cid:21)(cid:27) . By eqn. (28) ˜ τ α α = (cid:90) d q α d q α G ( q α , t α ; q α , t α ) G ( q α , t α ; q α , t α )= (cid:20) m π i (cid:126) ( t α − t α ) (cid:21) d/ (cid:20) m π i (cid:126) ( t α − t α ) (cid:21) d/ J α J α , (72)where the integrals J α and J α are defined as J α := (cid:90) d q α exp (cid:20) i m (cid:126) ( q α − q α ) t α − t α (cid:21) (73) × exp (cid:26) i (cid:126) (cid:20) F t α − t α )( q α + q α ) − F m ( t α − t α ) (cid:21)(cid:27) and J α := (cid:90) d q α exp (cid:20) i m (cid:126) ( q α − q α ) t α − t α (cid:21) (74) × exp (cid:26) i (cid:126) (cid:20) F t α − t α )( q α + q α ) − F m ( t α − t α ) (cid:21)(cid:27) . Now the integrals (73) and (74) are readily evaluated, with the results J α = (cid:20) π i (cid:126) ( t α − t α ) m (cid:21) d/ exp (cid:26) i (cid:126) (cid:20) − F m ( t α − t α ) + F q α ( t α − t α ) (cid:21)(cid:27) (75)and J α = (cid:20) π i (cid:126) ( t α − t α ) m (cid:21) d/ exp (cid:26) i (cid:126) (cid:20) − F m ( t α − t α ) + F q α ( t α − t α ) (cid:21)(cid:27) . (76)Finally substituting the integrals (75) and (76) into eqn. (72) we obtain the trivialisation τ α α = exp (cid:26) i (cid:126) (cid:20) − F m ( t α − t α ) + F q α ( t α − t α ) (cid:21)(cid:27) × exp (cid:26) i (cid:126) (cid:20) − F m ( t α − t α ) + F q α ( t α − t α ) (cid:21)(cid:27) . (77)Eqn. (77) correctly reduces to the free–particle trivialisation (70) when F = 0 .16 .3 The quadratic potential As a final example we will work out the trivialisation for an isotropic harmonic oscil-lator with frequency ω . Here the propagator is given by G ( q α , t α ; q α , t α ) = (cid:26) mω π i (cid:126) sin[ ω ( t α − t α )] (cid:27) d/ × exp (cid:18) i mω (cid:126) sin[ ω ( t α − t α )] (cid:8) ( q α + q α ) cos[ ω ( t α − t α )] − q α q α (cid:9)(cid:19) . (78)Again by eqn. (28) ˜ τ α α = (cid:90) d q α d q α G ( q α , t α ; q α , t α ) G ( q α , t α ; q α , t α )= (cid:26) mω π i (cid:126) sin[ ω ( t α − t α )] (cid:27) d/ (cid:26) mω π i (cid:126) sin[ ω ( t α − t α )] (cid:27) d/ K α K α , (79)where the integrals K α and K α are defined by K α := (80) (cid:90) d q α exp (cid:18) i mω (cid:126) sin[ ω ( t α − t α )] (cid:8) ( q α + q α ) cos[ ω ( t α − t α )] − q α q α (cid:9)(cid:19) and K α := (81) (cid:90) d q α exp (cid:18) i mω (cid:126) sin[ ω ( t α − t α )] (cid:8) ( q α + q α ) cos[ ω ( t α − t α )] − q α q α (cid:9)(cid:19) . One finds K α = (cid:26) π i (cid:126) mω tan [ ω ( t α − t α )] (cid:27) d/ exp (cid:26) − i mω (cid:126) tan [ ω ( t α − t α )] q α (cid:27) (82)and K α = (cid:26) π i (cid:126) mω tan [ ω ( t α − t α )] (cid:27) d/ exp (cid:26) − i mω (cid:126) tan [ ω ( t α − t α )] q α (cid:27) . (83)Substituting eqns. (82) and (83) into eqn. (79), normalising by the correspondingmodulus and dropping all constant phase factors we obtain the trivialisation τ α α = exp (cid:18) − i mω (cid:126) { tan [ ω ( t α − t α )] + tan [ ω ( t α − t α )] } q α (cid:19) . (84)Eqn. (84) also reduces to the free–particle trivialisation (70) when ω = 0 .17 igure The closed trajectory L α α α of eqn. (37). Acknowledgements
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