Volumes of Space as Subsystems
aa r X i v : . [ g r- q c ] N ov Volumes of Space as Subsystems
Federico Piazza ∗ Perimeter Institute for Theoretical PhysicsWaterloo, Ontario, N2L 2Y5, Canada
Fabio Costa
Perimeter Institute for Theoretical PhysicsWaterloo, Ontario, N2L 2Y5, Canada.Dipartimento di Fisica, Università di Milano
As a novel approach with possible relevance to semiclassical gravity, we propose to define regionsof space as quantum subsystems. After recalling how to divide a generic quantum system into“parts”, we apply this idea to a free scalar field in Minkowski space and we compare two differentlocalization schemes. The first scheme is the standard one, induced by the local relativistic fields;the alternative scheme that we consider is the one induced by the Newton-Wigner operators.If degrees of freedom are divided according to the latter, the Hamiltonian of the field exhibitsa certain amount of non-locality. Moreover, when a region of space is cut off from the restaccording to the Newton-Wigner scheme, the geometric entropy is finite and exhibits a sensiblethermodynamic behaviour.
From Quantum to Emergent Gravity: Theory and PhenomenologyJune 11-15 2007Trieste, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ olumes of Space as Subsystems
Federico Piazza
1. Subsystems and Local Subsystems
Among other famously intriguing and counterintuitive aspects of quantum physics, relativelylittle attention has been paid to the quantum mechanical description of a composite system. Thepartitions of a quantum system – i.e. all possible ways that a quantum system can be divided into“parts” – have a mathematical structure completely different from, for example, the partitions of aset in set theory. In set theory you can choose a bi-partition A-B by going through each element ofa (countable) set and deciding whether it belongs to subset A or B. Clearly, finite sets admit only afinite number of possible partitions. Analogously, a finite lattice can be divided into sub-volumesin a finite number of ways.Quantum mechanics divides things differently [1]; a quantum system can be partitioned if itsHilbert space can be written as a tensor product of Hilbert spaces. Consider, for instance, a systemdescribed by the Hilbert space C . The latter can be seen as a two-spin system and written as C = C A ⊗ C B , where indices A and B identify each of the two identical components C . Givenany orthonormal basis {| a i , | b i , | c i , | d i} of C , one way of partitioning the system is through theidentification | a i ≃ | i A ⊗ | i B , | b i ≃ | i A ⊗ | i B , | c i ≃ | i A ⊗ | i B , | d i ≃ | i A ⊗ | i B , (1.1)where {| i A , | i A } and {| i B , | i B } are some choosen basis in C A and C B respectively. A differentpartition is defined by the choice of another orthonormal basis, say {| a ′ i , | b ′ i , | c ′ i , | d ′ i} , to use forthe one to one correspondence (1.1). All possible partitions of C are thus given by the elements ofthe group SU ( ) except that, within SU ( ) , there are also transformations that merely correspondto a change of basis in either of the two factors C . These transformations have to be factored outsince they don’t change the partition, leaving us with the group SU ( ) / SU ( ) : those are all theinequivalent ways we can separate a two-spin system! Note therefore that quantum degrees offreedom, even when finite, can be split in an infinite number of ways. Not only can you choosewhether some of them belong to, say, subsystem A or B , but, as opposed to the elements of a setor the sites of a lattice, you can unitarily mix them before the splitting, in such a way that theycompletely lose their individual identities.Many appealing arguments in semi-classical gravity, such as those related to black hole ther-modynamics and to the holographic principle, are based upon the splitting of quantum degreesof freedom into two parts, each belonging to separate regions of space, typically across a causalhorizon or just across some imaginary boundary. According to the holographic principle [2], whengravity is taken into account, the total number of degrees of freedom is bounded by the area – ratherthan the volume – of the region. A breakdown of locality has also been invoked (e.g. [3]) in relationto the black hole information-loss paradox . Such hints are clearly in conflict with local quantumfield theory and call for a deep reassessment of our current physical understanding. Instead ofventuring into the highly arbitrary and unknown realm of possible non-local theories, here we takethe rather conservative point of view of maintaining the basic dynamics of our successful quantumtheories (e.g. the Hamiltonian of the Standard Model) and just allowing some flexibility when it More generally, a d N -dimensional Hilbert space can be partitioned into N smaller systems each of dimension d and such partitions are in one to one correspondence with the elements of SU ( d N ) / SU ( d ) N olumes of Space as Subsystems Federico Piazza comes to dividing their quantum degrees of freedom according to distinct regions of space. Such a“pre-geometric" approach [4] may look a bit fictitious since, after all, locality is already built intoquantum field theory and the correct “local” tensor product structure should turn out to be the oneactually induced by the local fields. In the rest of this paper we question this established point ofview by comparing a few aspects of the standard localization procedure with those of an alternativeone, induced by the “Newton-Wigner” [5] operators. Our main interest, rather than just philosoph-ical, is to present a different rationale to be possibly applied in semiclassical gravity whenever itcomes to isolating a bunch of “local” degrees of freedom. A more thoroughgoing and operationallybased case for alternative localization schemes will appear elsewhere [6].Although the general approach that we are following here is fairly recent [4], Newton Wigner(NW) operators are almost 60 years old. In this paper we review aspects of the standard andNW localization schemes that, to some extent, are already known in the literature, but in the newlight of [4]. At the end of section 4 we also mention some new results that will appear in moredetail elsewhere [7]. We will work in the Schroedinger picture where observables do not evolve intime. We will be rather cavalier about the mathematical subtleties involved with continuous tensorproducts: say that IR and UV regulators are implicitly assumed which make the total dimension of H finite.
2. The Two Localization Schemes
It is possible to assign a tensor product structure (TPS) to a system by specifying a set of acces-sible observables [8]. Consider a quantum system divided into two parts, P and R : H = H P ⊗ H R . P stands for “place” and R stands for “rest of the system”. Which tensor decomposition actuallydivides H into “places” is the matter of the present debate. If we have two sets of observables, A jP and A jR , separately defined in subsystem P and R respectively, then we can trivially extend suchobservables to the entire system as follows, A jP −→ A j ( P ) ≡ A jP ⊗ R , A jR −→ A j ( R ) ≡ P ⊗ A jR , (2.1)i.e. we just make them act as the identity on the other subsystem. By construction we have [ A j ( P ) , A k ( R )] = . (2.2)The basic idea here (see [8] for more detail and mathematical rigor) is that the converse is also true.That is if we isolate two subalgebras A ( P ) and A ( R ) , within the algebra of observables actingon H , satisfying (2.2), then they induce a unique bipartition H = H P ⊗ H R . Since in quantumfield theory (QFT) the usual local observables commute at space-like separated events, we have astraightforward realization of (2.2) and we can use local fields to define a local TPS at each time t .At the risk of being pedantic we will be more explicit. Consider a scalar field f , together withits conjugate momentum p , and a region of space P at some fixed time t in Minkowski spacetime.By a “localization procedure" we mean a rationale that relates the physical volume P to its quantumdegrees of freedom P by partitioning the total Hilbert space H of the field into H P ⊗ H R . If p is a Actually, only if the two subalgebras generate the entire algebra of operators on H [8] olumes of Space as Subsystems Federico Piazza point in P i.e. p ∈ P and r is not, i.e. r ∈ R , then from the usual commutation relations we clearlyhave that [ f ( p ) , f ( r )] = [ p ( p ) , p ( r )] = [ f ( p ) , p ( r )] = . (2.3)Also linear combinations of f , p and their spatial derivatives commute if they belong to the twoseparate regions P and R . In other words, relation (2.2) is satisfied if we take as the algebra ofoperators A ( P ) the one generated by the local fields in P . We call the corresponding partition the standard TPS or the standard localization scheme .Before introducing the Newton-Wigner localization scheme we first specify the Hamiltonian H of the field system. For simplicity we consider a free scalar field f of mass m : H = Z d k w k a † k a k , (2.4)where the usual infinite vacuum contribution has been subtracted, w k = √ k + m and operators a k satisfy the commutation relation [ a k , a k ′ ] = , [ a k , a † k ′ ] = d ( k − k ′ ) . The non self-adjoint Newton-Wigner fields a ( x ) are just defined as the Fourier transform of a k : a ( x ) = ( p ) / Z d k a k e i k · x , a † ( x ) = ( p ) / Z d k a † k e − i k · x . (2.5)On the other hand, in the definition of the relativistic fields f , the invariant relativistic measure ( w k ) − / appears in the integral, namely: f ( x ) = ( p ) / Z d k √ w k (cid:16) a k e i k · x + a † k e − i k · x (cid:17) . (2.6)Eq. (2.5) can be seen as a Bogoliubov transformation that doesn’t mix creators with annihilatorsand therefore doesn’t change the particle content of the system. As for any Bogoliubov transfor-mation the commutation relations are preserved, i.e. [ a ( x ) , a ( x ′ )] = , [ a ( x ) , a † ( x ′ )] = d ( x − x ′ ) .As before, if p ∈ P and r ∈ R (i.e. r / ∈ P ), we have [ a ( p ) , a ( r )] = [ a ( p ) , a † ( r )] = , (2.7)so that the subalgebras produced by the Newton Wigner fields also induce a TPS on H . (Also,for instance, the operators a k induce a TPS, but it goes without saying that such TPS is a “lesslocalized” one, being associated with modes of given momentum).
3. Some Properties of the two Schemes
Perhaps the most striking difference between these two schemes is that interactions are localin the standard localization scheme but not in the Newton-Wigner one. The Hamiltonian is in facta sum of pieces that are local only in the standard TPS: H = Z d x H ( x ) (3.1)4 olumes of Space as Subsystems Federico Piazza (here H ( x ) is the Hamiltonian density), but not in the Newton-Wigner one. For instance, in thecase of a free scalar field, from eqs. (2.4) and (2.5) we have H = Z d x d y K xy a † ( x ) a ( y ) . (3.2)The kernel inside the integral is a function of | x − y | that dies off as K xy ∼ e − m | x − y | for | x − y | ≫ m − and K xy (cid:181) | x − y | − in the massless case. Non locality is therefore exponentially suppressed atdistances larger than the Compton wavelength. For massless fields these effects are much moreserious although, in the more realistic case of several – massive and massless – fields interactingwith each other, it is not yet clear to us how to extend the NW localization.On the Compton wavelength scales one should also expect violations of causality. By switch-ing to the Heisenberg picture, not surprisingly, NW fields do not commute at spacelike separatedevents, they do commute only if they belong to the same hypersurface t = const . As opposed to f ( x , t ) , a ( x , t ) , as well as p ( x , t ) , are clearly not relativistically invariant objects, since their defini-tion depends on a foliation of spacetime into t = const hypersurfaces that has been choosen at thebeginning. A covariant extension of a ( x , t ) has therefore to include the hypersurface as a variable[9]. In other words, a has to be a function not only of ( x , t ) but also of the future-pointing, unit4-vector h m that locally represents the observer’s quadrivelocity. The Hilbert space of our field theory has a Fock structure: H = C ⊕ H ⊕ . . . ⊕ H n ⊕ . . . , (3.3)where H is the single particle space and the n -particles space, H n , is given by the symmetrictensor product of n copies of H . We have seen that a localization scheme is determined whena local algebra of operators A ( P ) , corresponding to a volume P , is specified. In A ( P ) , one canalways find ladder operators , that is, operators that take a vector of H j into one of H j + . Accord-ing to the NW scheme, these are just the NW operators a † ( p ) of eq. (2.5), with p ∈ P , and theirsuperpositions. In the standard formalism, on the other hand, one can consider the negative energypart of (2.6): f − ( p ) = ( p ) / Z d k √ w k a † k e − i k · p (3.4)and superpositions. By applying the ladder operators of A ( P ) and A ( R ) to the vacuum state,we find two linear varieties P and R in H , representing the one-particle excitations inside andoutside P according to some localization scheme. Accordingly, the single particle space H de-composes into a direct sum, H = P ⊕ R . (3.5)The key point here is that P and R are not necessarily orthogonal. They are in NW because ofthe commutation relations (2.7) but not in the standard localization scheme, since the two-pointfunction h | f ( x ) f ( x ′ ) | i (without T-product!) doesn’t vanish outside the lightcone. When P and R are orthogonal, one can make the identification P −→ P ⊗ | i R , R −→ | i P ⊗ R , (3.6)5 olumes of Space as Subsystems Federico Piazza which, rather intuitively, means that a particle well localized inside P leaves R “empty” and viceversa. This is not possible if P and R are not orthogonal because the RHSs of (3.6) are orthogonalby construction.In the NW scheme we can generalize the identification (3.6) and extend it to the whole Hilbertspace [10, 7]. The n -particles space, being a symmetric tensor product of copies of (3.5), decom-poses into H n = n M k = P k ⊗ R n − k , (3.7)where P k , R k represent symmetric tensor powers of P and R respectively. Again, the intuitiveinterpretation here is “if I have n particles they can be all in P and leave R empty, or I can have n − P and one particle in R , or n − H decomposesinto two Fock spaces H P and H R : H = ¥ M n = H n = ¥ M n = n M k = P k ⊗ R n − k = ¥ M n , m = P n ⊗ R m ≡ H P ⊗ H R . (3.8)This is not true in the standard localization scheme, where the corresponding H P and H R are not,independently, Fock spaces.To summarize, in the NW case P and R are orthogonal subspaces of H that correspondprecisely to the regions of space of first quantization . Thus, in the NW scheme we can fairlyinterpret each volume as a subsystem with an internal Fock structure compatible with the globalone. On the contrary, in the standard scheme the state of a particle localized in P is not orthogonal tothat of a particle localized in R ; as a consequence, we can still consider each volume as a subsystem,but not as a Fock space: particles are not separately defined in H P and H R (see also [11] on this).This is strictly related to the vacuum being entangled in the standard scheme. More on this in Sec.4. The usual scheme seriously challenges any idea of “localized state”. It sounds very natural todefine a state | y i as “strictly localized” [12] outside P if for any possible observable A in A ( P ) , h y | A | y i = h | A | i . In other words, if we excite some degrees of freedom that are “strictly local-ized” outside P , the state of affairs inside P is the same as the one of the vacuum, and I have nochance of detecting something different from the vacuum inside P by using the local operations A ( P ) . It turns out that no state with finite energy has this property in the standard localizationscheme. The state | y i ≡ f ( r ) | i which is commonly described as “a particle at position r ” is infact different from the vacuum in any region P with r / ∈ P , i.e. r P ≡ Tr R | y ih y | 6 = Tr R | ih | . Thisproperty, which can be traced back to Reeh-Schlieder theorem [20], is related, once again, with thefact that the vacuum is entangled in the standard scheme. On the other hand, low energy excitationscan be “strictly local” in the NW scheme because of the factorization (3.6) that leaves P empty andin its “local vacuum” whenever we excite some degrees of freedom somewhere else (i.e. in R ). Note that in the first quantization formalism regions of space are subspaces of H , rather than subsystems! olumes of Space as Subsystems Federico Piazza
4. Entropy
Although expressed as the integral of a local density, the energy (3.1) hides a certain amount ofnon-extensiveness. By isolating, as before, a region P from the rest R of Minkowski space, one eas-ily realizes that H = H ( P ) + H ( R ) . Just adding the inside and outside contribution, H ( P ) + H ( R ) ,in fact, leaves out of the Hamiltonian the UV-divergent contact term coming from the gradientsacross the boundary of P . It is because of such interaction terms that the vacuum is entangled inthe standard localization scheme: its Von Neumann entropy is UV-divergent and proportional tothe boundary of P [13].Von Neumann entropy is also known (see e.g. [14]) to be the appropriate generalization ofthermodynamical entropy for generic quantum states. In the case of conformal field theories theVon Neumann entropy of a region/subsystem has been calculated for a thermal state r (cid:181) e − b H in1+1 [15] and – using insights from AdS/CFT correspondence – also in higher dimensions [16].In such QFT calculations, in order to recover a thermodynamically sensible result (e.g. S therm ≃ V T for a massless field in 3 dimensions), the divergent contribution of the vacuum has to besystematically subtracted. Such a subtraction procedure, as noted in [17], is problematic becauseof the non-trivial dependence of the correction on area. Moreover, one can construct, starting fromthe vacuum, quite ad hoc states of higher and higher energy which are less and less entangled: afterthe subtraction those states would end up having a negative entropy!Clearly, a basic issue to be understood is whether or not such a divergent entropy actuallyaccounts for practically measurable correlations, i.e. whether or not it has any operational meaning.If the procedure described in [18] to create EPR pairs from vacuum entanglement turned out to beexperimentally practicable, this would strongly suggest that the standard localization scheme is thecorrect way to isolate local quantum degrees of freedom.In this respect, the NW localization scheme can be seen as a sort of UV-regulator. If we isolatea region of space according to the NW procedure we find in fact that the vacuum is a product state | i = | i P ⊗ | i R and the corresponding Von Neumann entropy is zero. In the free field case (2.4),if we switch the temperature on, the (non normalized) reduced density matrix r P (cid:181) Tr R e − b H isblock diagonal in each Fock subspace of given particle number. The trace of its n th power nicelyrearranges in an exponential, giving [7]Tr P r nP = exp ¥ (cid:229) j = j Tr K jn ! . (4.1)Here K is the two-point function K ( p , p ) ≡ P h | a ( p ) r P a † ( p ) | i PP h | r P | i P , (4.2)where a ( p ) and a † ( p ) are the Newton Wigner operators (2.5) and p and p are points inside P .The trace on the RHS of (4.1) is made inside subsystem P and limited to one-particle subspace:Tr K m ≡ Z p ... p m ∈ P d p d p . . . d p m K ( p , p ) K ( p , p ) . . . K ( p m , p ) . (4.3)7 olumes of Space as Subsystems Federico Piazza
From (4.1) we can then use the trick (see e.g. [17]) S ≡ − Tr P ( r P ln r P ) = (cid:18) − ddn + (cid:19) ln Tr P r nP (cid:12)(cid:12)(cid:12)(cid:12) n = (4.4)to calculate the Von Neumann entropy S . While referring to [7] for more details, here we point outthat the entropy (4.4) – in the NW localization scheme – is a thermodynamically sensible quantity:it doesn’t have UV divergences, it vanishes at zero temperature and gradually increases to reachthe S ∼ V T behavior (for a massless field) in the high temperature T ≫ V − / limit. At no stagehave we found an area dependent contribution.
5. Conclusions
We have considered two different localization schemes i.e. two different ways of relating somephysical volume P to its quantum degrees of freedom P by partitioning the total Hilbert space H of the field into H P ⊗ H R . We stress again that going from one tensor product structure H P ⊗ H R to another is not like playing with the points of space across the border of P , or choosing some dif-ferent smearing or compact support function for our definitions. As explained in the introduction,changing TPS is deeper than “playing with the parts” of a set in the usual intuitive sense: here P isa subset of R , P is a subsystem.As long as we are concerned only with the internal dynamics of the fields, all TPSs describeprecisely the same state of affairs: we are just considering different – equally valid – partitionsinto subsystems of the field system, not changing its dynamics (cross sections, decay rates etc. . . ).Things may possibly be different when also gravity is taken into account. In the standard approach,gravity is included in the action principle S = R d x ( R + L matter ) , solidly binding us to the standardlocalization scheme and to a local (gravity + matter) theory. Rather adventurously, one may insteadstick with the genuine and naive idea that (semiclassical) gravity is really just the geometry of thephysical spacetime. Then it would be crucial to understand how the matter degrees of freedomfeeding into Einstein equations are “localized” in the physical spacetime itself. By incorporat-ing alternative localization schemes, semiclassical gravity inherits from the matter fields a certainamount of non-locality (see eq. 3.2), although a consistent formulation of such a non local theory(gravity + matter) has yet to be written and surely calls for a major breakthrough. Acknowledgments
It is a pleasure to thank Michele Arzano, Sergio Cacciatori, Lucien Hardy, Justin Khoury,Matthew Leifer, Simone Speziale and Andrew Tolley for exciting discussions and help. F.C. wouldlike to thank Perimeter Institute for hospitality during the Undergraduate Research Project of sum-mer 2007. This research was supported by Perimeter Institute for Theoretical Physics. Research atPerimeter Institute is supported by the Government of Canada through Industry Canada and by theProvince of Ontario through the Ministry of Research and Innovation.
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