JMathPhys_39_5083

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 10 OCTOBER 1998 Downloaded 13 Ap Corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models Stephen L. Adlera) Institute for Advanced Study, Princeton, New Jersey 08540 Achim Kempf Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, United Kingdom ~Received 13 February 1998; accepted for publication 18 May 1998! We study the leading corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models, by deriving several related Ward identities, and give conditions for these corrections to be small. We show that emergent canonical commutators are possible only in matrix models in complex Hilbert space for which the numbers of fermionic and bosonic fundamental degrees of freedom are equal, suggesting that supersymmetry will play a crucial role. Our results simplify, and sharpen, those obtained earlier by Adler and Millard. © 1998 American Institute of Physics. @S0022-2488~98!03310-6# I. INTRODUCTION It is widely believed that at distances of order the Planck length l P;10233 cm our conven- tional notions of the geometry of space–time break down, as a result of quantum gravity effects. One indication of the modifications in physics that might be expected is provided by string theory models of quantum gravity, in which several studies suggest a modification of the uncertainty relation of the form1 DxDp> \ 2 @11b~Dp ! 21fl# , b.0, ~1a! implying a finite minimum uncertainty Dx05\b1/2 in the vicinity of the Planck length. As dis- cussed by Kempf and collaborators,2 Eq. ~1a! corresponds to a correction to the Heisenberg canonical commutation relations of the form @x ,p#5i\~11bp21fl !. ~1b! We wish in this paper to discuss modifications of the Heisenberg algebra arising in another context, that of the statistical mechanics of matrix models, and to compare them with Eqs. ~1a, 1b!. Several years ago, Adler proposed a set of rules for a generalized quantum or trace dynamics, which is a Lagrangian and Hamiltonian mechanics with arbitrary noncommutative phase space variables q, p, and this was developed in a series of papers with various collaborators.3 For theories in which the action is constructed as the trace of a sum of matrix products of N3N matrix variables, trace dynamics gives a powerful, basis independent, way of representing the same dynamics that can also be described in terms of the N2 individual matrix elements. A significant new result emerging from this point of view was obtained by Adler and Millard,4 who argued that the statistical mechanics of trace dynamics takes the form of conventional quantum field theory, with the Heisenberg commutation relations holding for statistical averages over certain effective canonical variables obtained by projection from the original operator canonical variables. Re- cently, it has become clear5 that the underlying assumptions of trace dynamics are satisfied by a!Author to whom correspondence should be addressed. Phone: 609-734-8051; fax: 609-924-8399; electronic mail: adler@ias.edu 50830022-2488/98/39(10)/5083/15/$15.00 © 1998 American Institute of Physics r 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Downloaded 13 Ap matrix models, for which the methods of trace dynamics provide a very convenient calculational tool. Hence the results of Adler and Millard can be reinterpreted as providing a statistical me- chanics of matrix models, and showing that thermal averages in this statistical mechanics can behave as Wightman functions in an emergent local quantum field theory. These results, together with recent work6 suggesting that the underlying dynamics for string theory may be a form of a matrix model, raise in turn the question of determining the form of the leading corrections to the Heisenberg algebra implied by the statistics of matrix models, formulating conditions for these corrections to be small, and seeing whether they can be related to the string theory result of Eqs. ~1a, 1b!. An investigation of these questions is the focus of this paper, which is organized as follows. In Sec. II we give a brief synopsis of the rules of trace dynamics in the context of matrix models. We show that the conservation of the operator4,7 C˜ can be understood as a simple consequence of unitary invariance. We also remark that, with Grassmann fermions, C˜ is independent of the classical parts of the matrix phase space variables, and review the statistical mechanics4,8 of matrix models. In Sec. III we consider the simple case of a bosonic matrix model with Hamiltonian quadratic in the canonical momenta, and, making no approximations, derive a simplified form of the Ward identity used in Ref. 4 to obtain the effective canonical algebra. This analysis shows that there are corrections to the canonical commutator quadratic in the canonical momentum. In Sec. IV we use the symplectic formalism of Ref. 4 to repeat this calculation in the case of a general commutator/anticommutator of canonical variables in a generic matrix model, that can include fermions, and we generalize the treatment of Ref. 4 to allow nonzero sources for the classical parts of the matrix variables. From the analyses of Secs. III and IV, we formulate conditions for the corrections to the emergent canonical algebra to be small. We show that these conditions require C˜ to be an intensive rather than extensive thermodynamic quantity, and that they can be satisfied in complex Hilbert space ~if at all! only in matrix models with precisely equal numbers of bosonic and fermionic degrees of freedom. This result strongly suggests that candidate matrix models for prequantum mechanics should be supersymmetric. We conclude by generalizing the conditions to ones that permit the recovery of the full emergent quantum field theory structure derived in Ref. 4. We also compare the prequantum corrections to the canonical algebra derived in Secs. III and IV, in which ~as is usual in field theories! the spatial coordinate is simply a label, and the field variables are the dynamical canonical variables, to the string theory inspired expression of Eq. ~1b! in which x is a coordinate operator. II. THE STATISTICAL MECHANICS OF MATRIX MODELS We begin by reviewing the statistical mechanics of trace dynamics, taking into account the simplifications5 that become possible when Grassmann algebras are employed to represent the fermion/boson distinction. Let B1 and B2 be two N3N matrices with matrix elements that are even grade elements of a complex Grassmann algebra, and Tr the ordinary matrix trace, which obeys the cyclic property Tr B1B25( m ,n ~B1!mn~B2!nm5( m ,n ~B2!nm~B1!mn5Tr B2B1 . ~2a! Similarly, let x1 and x2 be two N3N matrices with matrix elements that are odd grade elements of a complex Grassmann algebra, which anticommute rather than commute, so that the cyclic property for these takes the form Tr x1x25( m ,n ~x1!mn~x2!nm52( m ,n ~x2!nm~x1!mn52Tr x2x1 . ~2b! The cyclic/anticyclic properties of Eqs. ~2a, 2b! are just those assumed for the trace operation Tr of trace dynamics. @In Refs. 3 and 4 the fermionic operators were realized as ordinary matrices 5084 J. Math. Phys., Vol. 39, No. 10, October 1998 S. L. Adler and A. Kempf with complex matrix elements, all of which anticommute with a grading operator (21)F which formed part of the definition of the graded trace Tr, for which fermions then obeyed Eq. ~2b! while bosons obeyed Eq. ~2a!. Since the use of Grassmann odd fermions eliminates the need for r 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Downloaded 13 Ap the inclusion of the (21)F factor, the graded trace obeying Eqs. ~2a, 2b! is just the usual matrix trace, for which we use the customary notation Tr.# From Eqs. ~2a, 2b!, one immediately derives the trilinear cyclic identities Tr B1@B2 ,B3#5Tr B2@B3 ,B1#5Tr B3@B1 ,B2# , Tr B1$B2 ,B3%5Tr B2$B3 ,B1%5Tr B3$B1 ,B2%, Tr B$x1 ,x2%5Tr x1@x2 ,B#5Tr x2@x1 ,B# , Tr x1$B ,x2%5Tr$x1 ,B%x25Tr@x1 ,x2#B , ~2c! which are used repeatedly in trace dynamics calculations. The basic observation of trace dynamics is that given the trace of a polynomial P constructed from noncommuting matrix or operator variables ~we shall use the terms ‘‘matrix’’ and ‘‘opera- tor’’ interchangeably in the following discussion!, one can define a derivative of the c-number Tr P with respect to an operator variable O by varying and then cyclically permuting so that in each term the factor dO stands on the right, giving the fundamental definition d Tr P5Tr d Tr P dO dO , ~3a! or in the condensed notation that we shall use throughout this paper, in which P[Tr P , dP5Tr dP dO dO . ~3b! Letting L@$qr%,$q˙ r%# be a trace Lagrangian that is a function of the bosonic or fermionic operators $qr% and their time derivatives ~which are all assumed to obey the cyclic relations of Eqs. ~2a-2c! under the trace!, and requiring that the trace action S5*dtL be stationary with respect to varia- tions of the qr’s that preserve their bosonic or fermionic type, one finds3 the operator Euler- Lagrange equations dL dqr 2 d dt dL dq˙ r 50. ~3c! Because, by the definition of Eq. ~3a!, we have S dLdqrD i j5 ]L ]~qr! j i , ~3d! for each r the single Euler-Lagrange equation of Eq. ~3c! is equivalent to the N2 Euler-Lagrange equations obtained by regarding L as a function of the N2 matrix element variables (qr) j i . Defining the momentum operator pr conjugate to qr , which is of the same bosonic or fermionic type as qr , by pr[ dL dq˙ r , ~4a! the trace Hamiltonian H is defined by H5Tr( r prq˙ r2L. ~4b! 5085J. Math. Phys., Vol. 39, No. 10, October 1998 S. L. Adler and A. Kempf In correspondence with Eq. ~3d!, the matrix elements (pr) i j of the momentum operator pr just correspond to the momenta canonical to the matrix element variables (qr) j i . Performing general r 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Downloaded 13 Ap same-type operator variations, and using the Euler-Lagrange equations, we find from Eq. ~4b! that the trace Hamiltonian H is a trace functional of the operators $qr% and $pr%, H5H@$qr%,$pr%# , ~5a! with the operator derivatives dH dqr 52p˙ r , dH dpr 5erq˙ r , ~5b! with er51(21) according to whether qr ,pr are bosonic ~fermionic!. Letting A and B be two trace functions of the operators $qr% and $pr%, it is convenient to define the generalized Poisson bracket $A,B%5Tr( r erS dAdqr dBdpr2 dBdqr dAdprD . ~6a! Then using the Hamiltonian form of the equations of motion, one readily finds that for a general trace functional A@$qr%,$pr%# , the time derivative is given by d dt A5$A,H%; ~6b! in particular, letting A be the trace Hamiltonian H, and using the fact that the generalized Poisson bracket is antisymmetric in its arguments, it follows that the time derivative of H vanishes. An important property of the generalized Poisson bracket is that it satisfies3 the Jacobi identity, $A,$B,C%%1$C,$A,B%%1$B,$C,A%%50. ~6c! As a consequence, if Q1 and Q2 are two conserved charges, that is if 05 d dt Q15$Q1,H%, 05 d dt Q25$Q2,H%, ~6d! then their generalized Poisson bracket $Q1,Q2% also has a vanishing generalized Poisson bracket with H, and is conserved. This has the consequence that Lie algebras of symmetries can be represented as Lie algebras of trace functions under the generalized Poisson bracket operation. A significant feature of trace dynamics is that, as discovered by Millard,7 the anti-self-adjoint operator7,4 C˜ [ ( r bosons @qr ,pr#2 ( r fermions $qr ,pr% ~7! is conserved by the dynamics. As we shall now show, conservation of C˜ holds whenever the trace dynamics has a global unitary invariance, that is, whenever the trace Hamiltonian obeys H@$U†qrU%,$U†prU%#5H@$qr%,$pr%# ~8a! for a constant unitary N3N matrix U, or equivalently, by Eq. ~4b!, whenever the trace Lagrangian obeys L@$U†qrU%,$U†q˙ rU%#5L@$qr%,$q˙ r%# . ~8b! Letting U5exp L, with L an anti-self-adjoint bosonic generator matrix, and expanding to first 5086 J. Math. Phys., Vol. 39, No. 10, October 1998 S. L. Adler and A. Kempf order in L, Eq. ~8a! implies that H@$qr2@L ,qr#%,$pr2@L ,pr#%#5H@$qr%,$pr%# . ~9a! r 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Downloaded 13 Ap But applying the definition of the variation of a trace functional given in Eq. ~3b!, Eq. ~9a! becomes Tr( r S 2 dHdqr @L ,qr#2 dHdpr @L ,pr# D50, ~9b! which by use of the trilinear cyclic identities of Eq. ~2c! yields Tr L( r S dHdqr qr2erqr dHdqr 1 dHdpr pr2erpr dHdprD50. ~9c! Since the generator L is an arbitrary anti-self-adjoint N3N matrix, the anti-self-adjoint matrix that multiplies it in Eq. ~9c! must vanish, giving the matrix identity ( r S dHdqr qr2erqr dHdqr 1 dHdpr pr2erpr dHdprD50. ~10a! But now substituting the Hamilton equations of Eq. ~5b!, Eq. ~10a! takes the form 05( r ~2p˙ rqr1erqrp˙ r1erq˙ rpr2prq˙ r! 5 d dt (r ~2prqr1erqrpr! 5 d dt S (r bosons @qr ,pr#2 (r fermions $qr ,pr% D , ~10b! completing the demonstration of the conservation of C˜ . Corresponding to the fact that C˜ is conserved in any matrix model with a global unitary invariance, it is easy to see4,8 that C˜ can be used to construct the generator of global unitary transformations of the Hilbert space basis. Consider the trace functional GL52Tr LC˜ , ~11a! with L a fixed bosonic anti-self-adjoint operator, which can be rewritten, using cyclic invariance of the trace, as GL5Tr( r @L ,pr#qr52Tr( r pr@L ,qr# . ~11b! Hence for the variations of pr and qr induced by GL as canonical generator, which have a structure analogous to the Hamilton equations of Eq. ~5b!, we get dpr52 dGL dqr 52@L ,pr# , dqr5er dGL dpr 52@L ,qr# . ~11c! Comparing with Eqs. ~8a! and ~9a!, we see that these have just the form of an infinitesimal global unitary transformation. For each phase space variable qr ,pr , let us define the classical part qr c ,pr c and the noncom- mutative remainder qr8 ,pr8 , by qr c5 1 N Tr qr , pr c5 1 N Tr pr , qr85qr2qr c , pr85pr2pr c , ~12a! 5087J. Math. Phys., Vol. 39, No. 10, October 1998 S. L. Adler and A. Kempf so that bosonic qr c ,pr c are c-numbers, fermionic qr c ,pr c are Grassmann c-numbers ~where by a c-number we mean a multiple of 1N , the N3N unit matrix!, and the remainders are traceless, r 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Downloaded 13 Ap Tr qr85Tr pr850. ~12b! Then since qr c ,pr c commute ~anticommute! with qs8 ,ps8 for r, s both bosonic ~fermionic!, we see that the classical parts of the phase space variables make no contribution to C˜ , and Eq. ~7! can be rewritten as C˜ 5 ( r bosons @qr8 ,pr8#2 ( r fermions $qr8 ,pr8%. ~12c! Thus C˜ is completely independent of the values of the classical parts of the matrix phase space variables. Making the assumption that trace dynamics is ergodic ~which undoubtedly requires an inter- acting as opposed to a free theory, and may presuppose taking the N!` limit!, one can then analyze4 the statistical mechanics of trace dynamics for the generic case in which the conserved quantities are the trace Hamiltonian H and the operator C˜ . As discussed in detail in the second paper cited in Ref. 5, the analysis of Ref. 4 carries over to the case in which the fermions are represented by Grassmann matrices; the demonstration of a generalized Liouville theorem still holds, and the requirements for convergence of the partition function are much less stringent, eliminating the complexities addressed in Appendix F of Ref. 4. With Grassmann fermions, for the typical models we are studying the bosonic part of H is a positive operator, from which H inherits good positivity properties. The canonical ensemble then takes the simple form given in Eq. ~48c! of Ref. 4, r5Z21 exp~2tH2Tr l˜C˜ !, Z5E dm exp~2tH2Tr l˜C˜ !, ~13! with dm the invariant matrix ~or operator! phase space measure provided by Liouville’s theorem, with t a real number, and with l˜ an anti-self-adjoint matrix that in the generic case ~which we assume! has no zero eigenvalues. ~Equation ~13! can be derived directly4 by maximizing the entropy subject to the constraints imposed by the conservation of H and C˜ , or indirectly8 by first calculating the corresponding microcanonical ensemble corresponding to these conserved quanti- ties, and then using standard statistical physics methods to calculate the canonical ensemble from the microcanonical one.! We wish to make two points about the partition function defined in Eq. ~13!. First, it is not invariant under the unitary transformation of Eq. ~8a! for fixed l˜, but is invari- ant when l˜ is simultaneously transformed to U†l˜U; hence the partition function breaks unitary invariance, but has a specific form of unitary covariance. Second, the partition function contains a weighted sum over all possible commutators @qr ,ps# for bosonic variables and all possible anti- commutators $qr ,ps% for fermionic variables; there is no restriction to the classical or quantum mechanical evaluation of these commutators/anticommutators as 0 or idrs respectively. However, statistical integrals like Eq. ~13! are typically dominated by specific regions of the integration domain, and we will see, by a study of the Ward identities following from Eq. ~13!, that this can lead to effective quantum mechanical commutators inside statistical averages. The structure of the Ward identities or equipartition theorems following from Eq. ~13! will be reanalyzed in the next two sections without making approximations used in Ref. 4, so as to determine the leading corrections to the emergent canonical commutation relations. From this analysis we will infer a set of conditions for obtaining the full emergent quantum field theory structure of Ref. 4. III. CORRECTIONS TO THE BOSONIC COMMUTATOR †qs ,pr‡ IN A SIMPLIFIED UNITARY INVARIANT MATRIX MODEL We consider in this section the simplified bosonic matrix model with trace Hamiltonian H5TrF( r 1 2 pr 21V~$qr%!G , ~14a! 5088 J. Math. Phys., Vol. 39, No. 10, October 1998 S. L. Adler and A. Kempf with the qr self-adjoint N3N complex matrix variables and with V a global unitary invariant potential. This form is general enough to include the matrix model forms of the bosonic field r 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Downloaded 13 Ap theories of greatest interest, including the Goldstone model, non-Abelian gauge models, and the Higgs model. As we saw in the previous section, the Hamiltonian dynamics for this model conserves both the real number H and the matrix C˜ , which in this case is given simply by C˜ 5( r @qr ,pr# . ~14b! Letting r and Z be respectively the canonical ensemble and partition function given in terms of H and C˜ by Eq. ~13!, we define the ensemble average of an arbitrary function O of the dynamical variables by ^O &AV5E dmrO 5Z21E dme2tH2Tr~l˜ C˜ !O . ~15! Letting O be the conserved operator C˜ , and noting that the right hand side of Eq. ~13! can be a function only of the ensemble parameters l˜ and t, we have ^C˜ &AV5 f ~l˜,t!, ~16a! with f an anti-self-adjoint matrix, which in general can be written as a phase matrix ieff times a commuting magnitude matrix u f u, f 5ieffu f u, ieff2 521, ieff† 52ieff , @ ieff ,u f u#50. ~16b! We shall now specialize to an ensemble for which the magnitude matrix u f u makes no dis- tinction among the different bases in Hilbert space, and so takes the form of a positive real multiple