Exotic Statistics for Ordinary Particles in Quantum Gravity
aa r X i v : . [ g r- q c ] M a y Exotic Statistics for Ordinary Particles in Quantum Gravity
John SwainMarch 31, 2008
Department of Physics, Northeastern University, Boston, MA 02115, USAemail: [email protected] (Awarded an Honorable Mention in the Gravity Research Foundation Essay Competition, 2008)
SUMMARY
Objects exhibiting statistics other than the familiar Bose and Fermi onesare natural in theories with topologically nontrivial objects including geons,strings, and black holes. It is argued here from several viewpoints that thestatistics of ordinary particles with which we are already familiar are likely tobe modified due to quantum gravity effects. In particular, such modificationsare argued to be present in loop quantum gravity and in any theory whichrepresents spacetime in a fundamentally piecewise-linear fashion. The ap-pearance of unusual statistics may be a generic feature (such as the deformedposition-momentum uncertainty relations and the appearance of a fundamen-tal length scale) which are to be expected in any theory of quantum gravity,and which could be testable. . INTRODUCTION The spin-statistics theorem [1] states that half-integer spin particle obey Fermi-Diracstatistics and the Pauli exclusion principle and integer spin particles obey Bose-Einsteinstatistics is well-known and has many derivations [2]. I argue in this essay that lifting therestrictions in the usual derivations of this theorem leads one to expect generically that itwill not hold when quantum gravitational effects are included. In fact, I will give so manyarguments for this that I argue that this may well be a generic feature of quantum theoriesof gravity [3]. Conversely, if quantum gravity effects leave the usual field commutators andspin-statistics connection untouched, then something very deep must be going on to protectthem.
II. THE STANDARD APPROACHES TO THE SPIN-STATISTICS THEOREM
There are two main ways to approach the spin-statics theorem. The first is to considercommutation relations for fields φ ( t, x ) and their conjugate momenta π ( t, x ) and assume thatit makes sense to think of them as being analogous to x and p of nonrelativistic quantummechanics. Suppressing Lorentz indices, one has expressions of the form[ φ ( t, x ) , φ ( t, x ′ )] = 0 (1)[ π ( t, x ) , π ( t, x ′ )] = 0 (2)[ φ ( t, x ) , π ( t, x ′ )] = iδ (3) ( x − x ′ ) (3)where the brackets can represent commutators or anticommutators. Ignoring interactionsand assuming locality and flat Minkowski spacetime, one then finds that one runs into2roblems if commutators are used for Dirac (half integer spin) fields or if anticommutatorsare used for Bose (integer spin) fields.An alternative viewpoint makes the swap of x and x ′ more physical by arguing [4] thatit is equivalent to a rotation (FIG. 1). x x’ FIG. 1. Two particles (small circles) swapped by a rotation (two arcs with arrows). Here noflux passes through the circle that defines the rotation, but if it did, one might expect additionalphases to appear.
If one thinks of two particles being swapped as equivalent to a rotation of 2 π , then onepicks up a sign of e ( i πs ) for spin s : a “+” for bosons and a “ − ” for fermions. Implicit in suchan argument is that there are no gauge fields present which could have altered the phasesof particles en route . III. EFFECTS OF QUANTUM GRAVITY
The most obvious concern with equations (1), (2), and (3) above is that they are equal-time commutation relations with an obvious reliance on a 3+1 split – something which mightbe dealt with by considering Peierls brackets instead [7]. Of course one would still have to3orry about possible extensions (central or not) when replacing brackets with commutatorsin the quantum theory.Rather general physical arguments suggest that the commutators between x and p shouldbe modified by quantum gravitational effects [5], as should the x commutators [6]. . Themomentum commutators are already nontrivial in curved spacetime.For fields, the δ (3) ( x − x ′ ) of (3) might be replaced by a sharply-peaked function withwidth related to the Planck mass. It is worth noting that such scale-dependent effects arisenaturally for composite particles. The idea that statistics of composite particles might besubtle goes back to Wigner [9] in 1929 and Ehrenfest and Oppenheimer [10] in 1931. Forfermion pairs (“quasibosons” [11]) such as superfluid helium-4 and Cooper pairs a short-range Pauli effect is present. Lipkin [12] notes that the energy gap for Cooper pairs “...wouldbe absent if the fermions behaved like simple bosons.” In other words, we already know ofsystems which behave more or less bosonic or fermionic as a function of scale.The most convincing general arguments for changes to the basic commutators are per-haps those of Ahluwalia-Khalilova [13] based on the work of Mendes, Chryssomalakos andOkon [14]. The idea is to find the most general stable extension of the combined Poincar´eand Heisenberg algebras. This leads to the Snyder-Yang-Mendes algebra [15], which hasnontrivial commutators (not all commutation relations are shown) between coordinates X µ and momenta P µ : [ X µ , X ν ] = iℓ P J µν (4) I would argue for nontrivial commutation relation for positions already without quantum gravity.[8] P µ , P ν ] = i ~ ℓ C J µν (5)[ P µ , X ν ] = i ~ η µν F + i ~ βJ µν (6)These involve ~ (explicit here), two length scales ℓ P (presumably of order the Planck length)and ℓ C (presumably of cosmological size), a new dimensionless constant β , and the angularmomentum J µν . F is a new operator having nontrivial commutation relations with P and X . The triply special relativity of Kowalski-Glikman and Smolin [16] can be put in thisform [13].Given that one postulates field commutators by analogy with the commutation relationsof x and p , it is now by no means obvious that (1),(2) and (3) are correct.One can also stick to the usual commutation relations for the Poincar´e and Heisenberggroup and ask questions at the level of the fields themselves. After all, physically (and inthe spirit of noncommutative geometry [17]), one constructs spacetime from measurementsinvolving fields.Now consider the product φ ( x ) φ ( y ) which is needed to form commutators, anticommu-tators and propagators ( x and y now commuting labels for spacetime points). The firstproblem is that φ ( x ) φ ( y ) is not gauge invariant. As noted long ago by Schwinger, φ ( x ) φ ( y )would need to be multiplied by a phase exp( i R yx A µ dx µ ) for whatever connection A is rel-evant. For a self-interacting charged particle in flat spacetime one finds [18] that the freeinfrared propagator 1 / ( p − m ) is raised to a fractional power (1 + απ ) and becomes non-local (a charged particle carries a long-range field). It has been argued that Newtoniangravitational self-interaction even in flat spacetime will give a similar sort of correction [18].In a general curved space background [19], but ignoring self-interaction, DeWitt [20]5as given an exact representation for the Feynman propagator which is complicated andnonlocal . Of course all these discussions have assumed adiabatic processes where particlenumbers do not change as in the Unruh and Hawking effects [19].Since what originally looked like a well-defined product (or expectation value thereof)of two free fields in flat spacetime must be replaced for interacting fields in curved space-time by a nonlocal object, it seems unlikely that their (anti-)commutators would suffer nomodifications.The foregoing two arguments, one kinematical and the other dynamical, suggest thatthe usual commutation relations for quantum fields (and thus the usual derivations of thespin-statistics connection) may not be generally valid. There have been arguments in the lit-erature that there should be no unusual spin-statistics relationships in curved spacetime [22].They require assumptions about the (anti-)commutation relations, as well as the existenceof suitable flat regions of spacetime. Bardek et al. [23] considered exotic statistics in curvedspacetime and argued for their consistency and constancy in an expanding universe, whileScipioni has argued [24] that transitions of statistics might occur under some conditions. Acomprehensive list of references can be found in [25].At first glance, statistics describes exchanges of objects (at the particle level) and thus isabout representations of the permutation group of n objects S n . If an exchange is equivalentto a rotation [4], and one wants two exchanges to be the identity, then one is naturally ledto the representation of a single exchange by multiplying a wavefunction by ±
1. This wasformalized with path-integral arguments by Laidlaw and deWitt [26] who ruled out any Even then there are subtleties.Toms has argued [21] that there is an ambiguity due to the choiceof path integral measure. a k a † l − qa † l a k = δ kl .A theory of objects satisfying these deformed commutation relations has a Hamiltonianwhich is nonlocal and nonpolynomial in the field operators – something obviously suggestiveof curved spacetime and the expected nonlocalities described above.If the particles have internal degrees of freedom ( i.e. their wavefunctions are sectionsof C N , or something else) there can be inequivalent quantizations, labelled by irreduciblerepresentations of the braid group B n ( M ). All sorts of novel statistics are then possible[32,33]. Intuitively, internal degrees of freedom can keep track of how many times particleshave gone “around each other”.In 2+1 dimensions, it is well-known that particles coupled to gauge fields can formcomposite objects (“anyons”) with unusual statistics [34]. The physical idea is easy tounderstand. If one takes a particle around a flux Φ, one picks up an Aharonov-Bohm typephase exp( ig Φ) where g represents the coupling of the particle to the field. The braid group B n keeps track of how the flux lines twist around each other as the particles move in 2dimensions. Gravitational anyons in 2+1 dimensions [35] could be physically relevant at thesurfaces of black holes (perhaps even microscopic or virtual ones [36]).It has been suggested [38] that exotic statistics for charged particles ( i.e. some electronsin a white dwarf acting as bosons) could arise for particles whose angular momentum comes7artly from coupling to external electromagnetic fields. Similar effects might be anticipatedfrom gravitational fields.Strings open up completely new possibilities. For a space M like “ R with n pointsremoved” (say by black holes or some sort of spacetime foam) one could have π ( M ) = 0but strings would probe the loop space Ω M , with π (Ω M ) = π ( M ) = 0.Exotic statistics are also possible [39] for strings even in topologically trivial 3-dimensional manifolds, due to their ability to be linked and tangled. For this, a furthergeneralization of braid statistics is needed involving the “loop braid group” LB n [40]. Lestone imagine that such issues are purely academic, Niemi has argued for exotic statistics in“leapfrogging” vortex rings [41] in quantum liquids and gases which, like anyons, really exist in the physical world. Exotic statistics for strings in 4-D BF theory have also been discussedby Baez, Wise, and Crans [40]. Particles corresponding to states of strings might inheritexotic statistics which could test stringy models of particles. That said, stringiness can alsobe somewhat hidden in the nonlocality of theories of point particles coupled to long-rangefields [42].Gambini and Setaro [43] found fractional statistics for composites of charged one-dimensional objects and vortices. Fort and Gambini found Fermi-Bose transmutation [44]for point scalars and Nielsen-Olesen strings in the Maxwell-Higgs system, and fractionalstatistics [45] in a 3+1 dimensional system composed of an open magnetic vortex and anelectrical point charge. 8 V. EXOTIC STATISTICS FOR ORDINARY PARTICLES IN LQG
I now want to concentrate on a very physical reason why loop quantum gravity (LQG)[46] and any 3+1 formulation of a piecewise-linear (PL) [47] Regge-like [48,49] theory ofgravity naturally supports exotic statistics.The idea is very simple: in 3+1 dimensional Regge-type theories [50], the curvature isdistributional, with support on edges where flat 3-simplices meet. A particle picks up aphases as it moves around a line of singular curvature and that phase can contribute toexotic statistics.In LQG, spin-networks define natural dual PL simplicial geometries [52,51] of flat simpli-cial complexes with distributional curvature along 1-dimensional subspaces where they join.A spin network is a graph composed of lines carrying SU (2) representation labels which giveareas to surfaces they pierce. They meet at vertices labelled by intertwiners. If there are atleast 4 edges which meet at a vertex, one can think of the vertex as enclosed by faces whichget their areas from the edges that pierce them. The intertwiner determines the volumeenclosed (see FIG. 2 for an example). Curvature is represented by the deficit angle alongedges where simplices meet (see FIG. 3 for an example).If the distributional lines of curvature are taken to represent matter, we have physicalstrings and all the arguments for the loop braid group also appear in this context.9 jjj
21 34
FIG. 2. A portion of a spin network and an associated simplex. The areas of the faces of thesimplex are determined by the SU(2) representations j i on the edges which pierce them and thevolume by the intertwiner associated with the vertex.FIG. 3. View of 2-simplices fitting together with a deficit angle (the edges there should bebrought together to construct the curved 2-geometry with curvature at the central point). The3-dimensional picture is analogous, but the 5 triangles are now replaced by 5 tetrahedra from aspin-network and the point where the curvature is becomes a line of curvature. One could think ofthis figure as looking “down” on the 5 flat triangular faces of 5 flat tetrahedra joined along a lineof singular curvature perpendicular to the page. generic possi-bility in theories of quantum gravity and worthy of theoretical and experimental study. It isa natural extension of the now familiar ideas of looking for changes to the basic commutatorsof x and p . Such modifications might only appear at very large or very small scales ( ℓ C or ℓ P ) and could easily have avoided experimental limits so far [53].I would like to conclude with some speculations connecting ideas discussed here withother topics of current research. If one thinks of local supersymmetry transformations aslocal changes of statistics of objects which can be fermions or bosons, then local changesof symmetry lead naturally to local translation invariance and thus to general coordinateinvariance. Is there a connection here? Jackson [54] has argued that a suitable position-momentum commutator can describe many features of gravity. To make this plausible, recallthat essentially all the interesting things about the geometry of phase space in quantummechanics come from the one nontrivial commutator.Braiding and exotic statistics may also have some bearing on interpretations of StandardModel particles in terms of framed spin networks [55,56]. Framings arise naturally in spin-networks with q-deformed groups [37] which are needed in LQG with a cosmological constant(nontrivial [ P µ , P ν ]). Also of interest is [57] in which spin and statistics for spacetime and“internal” exchanges are connected. It is also interesting that non-commutative geometriesarise naturally together with q-deformed groups in the same situations where anyons appear[58], so it seems many ideas may be connected.11 . ACKNOWLEDGEMENTS I would like to thank Ka´ca Bradonji´c and Tom Paul for careful readings of drafts of thispaper. I would also like to thank all Dharam Ahluwalia for having brought the issue ofstability of a deformed Poincar´e-Heisenberg algebra to my attention, and various membersof the loop quantum gravity community for having said interesting things at one pointor another at LOOPS ’07, especially Lee Smolin, Fotini Markopoulou, Seth Major andSundance Bilson-Thompson. This work was supported in part by the US National ScienceFoundation.
Note added:
The day after this work was completed, a preprint from Mark G. Jacksonentitled “Spin-Statistics Violations from Heterotic String Worldsheet Instantons” appearedon Arxiv (ArXiv:0803.4472v1) which discusses possible violations of the spin-statistics the-orem in heterotic string theory and also makes the point that should the true scale forquantum gravity be much lower than the usual Planck scale, such effects might be morereadily observable in the near future. [1] R. F. Streater and A. S. Wightman, “PCT, Spin and Statistics, and All That”, PrincetonUniversity Press, 2000.[2] I. Duck and E. C. G. Sudarshan, “Pauli and the Spin-Statistics Theorem”, World Scientific,March 1998.[3] L. Smolin, “Generic Predictions of Quantum Theories of Gravity”, hep-th/0605052[4] M. V. Berry and J. M. Robbins, Proc. Roy. Soc.
A453 (1997) 1771.
5] There is a large literature on generalizing the uncertainty principle, which in a string theoryconcept goes back at least to Veneziano: G. Veneziano, Europhys. Lett. (3) (1986) 199. Fora review, see F. Scardigli, Phys. Lett. B452 (1999) 39.[6] D. V. Ahluwalia, Physics Letters
B339 (1994) 301 (Honorable Mention in the 1994 Awardsfor Essays in Gravitation)[7] See for example, G. Bimonte, G. Esposito, G. Marmo, and C. Stornaiolo, Int. J. Mod. Phys.
A18 (2003) 2033 and references therein.[8] S. Sivasubramanian, G. Castellani, N. Fabiano, A. Widom, J. Swain, Y.N. Srivastava, G.Vitiello, Annals Phys. (2004) 191-203; S. Sivasubramanian, G. Castellani, N. Fabiano, A.Widom, J. Swain, Y. N. Srivastava, and G. Vitiello, J. Mod. Optics, (2004) 1529.[9] E. P. Wigner, Math. und Naturwiss. Anzeiger der Ungar. Ak. der Wiss. (1929) 576 (citedin [11])[10] P. Ehrenfest and J. R. Oppenheimer, Phys. Rev. (1931) 333.[11] W. A. Perkins, Int. J. Theor. Phys. (2002) 823.[12] H. J. Lipkin, “Quantum Mechanics”, North-Holland, Amsterdam, 1973, Ch. 6 (quoted in [11]).[13] D. V. Ahluwalia-Khalilova, Class. Q. Grav. (2005) 1433.[14] R. V. Mendes, J. Phys. A27 (8091); C. Chryssomalakos and E. Okon, Int. J. Mod. Phys.
D13 (2004) 2003.[15] H. S. Snyder, Phys. Rev. (1947) 38; C. N. Yang, Phys. Rev. (1974) 874; R. V. Mendes,J. Phys. A27 (1994) 8091; R. V. Mendes, J. Math. Phys. (2000) 156;
16] J. Kowalski-Glikman and L. Smolin, Phys. Rev.
D70 (2004) 065020.[17] A. Connes, “Noncommutative Geometry”, Academic Press, 1994.[18] S. Gulzari, J. Swain, and A. Widom, Mod. Phys. Lett. (2006) 2861-2871; S. Gulzari, Y. N.Srivastava, J. Swain, and A. Widom, Proceedings of IRQCD, June 5–9, 2006, Rio de Janeiro,and Braz. J. Phys. vol. 37, no. 1b. March 2007, page 286.[19] N. D. Birrell and P. C. W. Davies,“Quantum fields in curved space”, Cambridge UniversityPress, 1982.[20] B. S. De Witt, “The Dynamical Theory of Groups and Fields”, in Relativity, Groups andTopology , eds. B. S. DeWitt and C. DeWitt, Gordon and Breach, 1965; Phys. Rep. (1975) 297.[21] D. J. Toms, “The Schwinger Action Principle and the Feynman Path Integral for QuantumMechanics in Curved Space”, hep-th/0411233.[22] L. Parker and Y. Wang, Phys. Rev.
D39 (1989); J. W. Goodson and D. J. Toms, Phys. Rev.Lett. (1993) 3240; R. Verch, Commun. Math. Phys. (2001) 261.[23] V. Bardek, S. Meljanac, and A. Perica, Phys. Lett. B338 (1994) 20.[24] R. Scipioni, Il Nuov. Cim., (1997) 119.[25] http://physics.nist.gov/MajResFac/EBIT/peprefs.html [26] M. G. G. Laidlaw and C. Morette DeWitt, Phys. Rev. D3 (1971) 1375.[27] J. L. Friedman and R.Sorkin, Phys. Rev. Le (1980) 1100; (1980) 148, General Relativityand Gravitation (1982) 615.
28] A. P. Balachandran, E. Batista, I. P. Costa e Silva and P. Teotonio-Sobrinho, Nuclear Physics
B566 (2000) 441;C. Anezeris, A. P. Balachandran, M. Bourdeau, S. Jo, T. R. Ramadas, and R. D. Sorkin,Mod. Phys. Lett. A4 (1989) 331; C. Aneziris, A. P. Balachandran, M. Bourdeau, S. Jo, R. D.Sorkin, and T. R. Ramadas Int. J. Mod. Phys. A4 (1989) 5459.[29] A. Strominger, Phys. Rev. Lett., (1993) 3397.[30] S. Doplicher, R. Haag, and J. Roberts, Comm. Math. Phys. (1971) 199: (1974) 49[31] O. W. Greenberg, Phys. Rev. Lett. (1990) 705; Phys. Rev. D43 (1991) 4111.[32] T. D. Imbo and E. C. G. Sudarshan, Phys. Rev. Lett. (1988) 481.[33] T. D. Imbo, C. S. Imbo, E. C. G. Sudarshan, Phys. Lett. B234 (1990) 103.[34] F. Wilczek, Phys. Rev. Lett. (1982) 1144; Phys. Rev. Lett. (1982) 957; R, Mackenzieand F. Wilczek, Int. J. Mod. Phys. A3 (1988) 2827.[35] S. Deser, Phys. Rev. Lett. , Y. M. Cho, D. H. Park, and C. G. Han, Phys. Rev. D43 (1991) 1421.[36] L. Crane and L. Smolin, Gen. Rel. Grav., (1985) 1209.[37] S. Major and L Smolin, Nucl. Phys. B473 (1996) 267.[38] A. Kato, G. Mu˜noz, D. Singleton, J. Dryzek, and V. Dzhunushaliev, Found. Phys. (2003)769; S. Mandal and S.Chakrabarty, “Electrons as quasi-bosons in Strong Magnetic Fieldsand the Stability of Magnetars”, astro-ph/0209462; J. Dryzek, A. Kato, D. Mu˜noz, and D.Singleton, “Electrons as quasi-bosons in magnetic white dwarfs”, astro-ph/0110320, D. Single- on and J. Dryzek, “Electromagnetic field angular momentum in condensed matter systems”,cond-mat/0009068.[39] J. A. Harvey and J. Liu, Phys. Lett. B240 (1990) 369; X. Fustero, R. Gambini, and A. Trias,Phys. Rev. Lett. (1989) 1964; C. Aneziris, A. P. Balachandran, L. Kauffman, and A. M.Srivastava, Int. J. Mod. Phys. A6 (1991) 2519, C. Aneziris, Mod. Phys. Lett. A7 (1992) 3789;S. Surya, J. Math. Phys. (2004) 2515.[40] J. C. Baez, D. K. Wise, and A. S. Crans, “Exotic Statistics for Strings in 4d BF theory”,gr-qc/0603085, 9 May 2006, published in Advances in Theoretical and Mathematical Physics,vol. 11., No. 5, October 2007[41] A. J. Niemi, Phys. Rev. Lett. (2005) 124502[42] P.-M. Ho, Phys.Lett. B558 (2003) 238.[43] R. Gambini and L. Setaro, Phys. Rev. Lett. (1990) 2623.[44] H. Fort and R. Gambini, Phys. Lett. B372 (1996) 226.[45] H. Fort and R. Gambini, Phys. Rev.
D54 (1996) 1778.[46] See for example, C. Rovelli,“Quantum Gravity”, Cambridge University Press, 2004 and T.Thiemann, “Modern Canonical Quantum General Relativity”, Cambridge University Press,2007.[47] T. Regge and R. M. Williams, J. Math. Phys. (2000) 3964.[48] T. Regge, Nuov. Cim. (1961) 558.[49] J. Ambjørn, M. Carfora, and A. Marzuoli, “The Geometry of Dynamical Triangulations”, ecture Notes in Physics, Springer, Berlin, 1997.[50] T. Piran and R. M. Williams, Phys. Rev. D33 (1986) 1622.[51] J.Swain, talk at Loops ’07 and paper in preparation.[52] F. Markopoulou, gr-qc/9704013.[53] O. W. Greenberg and R. N. Mohapatra, Phys. Rev.
D39 (1989) 2032.[54] M. G. Jackson, Int. J. Mod. Phys.
D14 (2005) 2239 (Honorable Mention in the 2005 Awardsfor Essays in Gravitation)[55] S. O. Bilson-Thompson, hep-ph/0503213[56] S. O. Bilson-Thompson, F. Markopoulou and L. Smolin, Class.Quant.Grav. (2007) 3975-3994.[57] J. Anandan, Phys. Lett. A248 (1998) 124.[58] E. G. Floratos, “Bohm-Aharonov Interactions and q-quantum mechanics”, Proceedings of theLepton-Photon Symposium, 1991, vol. 1, 107.17