the principle of locality and quantum field theory on

THE PRINCIPLE OF LOCALITY AND QUANTUM FIELD THEORY ON (NON GLOBALLY H YPERBOLIC) CURVED SPACETIMES Dedicated to Rudolf Haag on the occasion of his 70th birthday BERNARD S. KAY DAMTP, University of CambridJJe, 3 Silver Street, Cambridge CB3 9EW, UK

Received 10 June 1992

In the contel.,J-1.

(i.e. linearity in test

(iii) for all test functions F (i.e. the equation of motion holds in a weak sense.) Note that this property is justified by the linearity of our model equation and (iv)

(3.1)

where .6. denotes the advanced minus retarded fundamental solution discussed in Sec. 2. (This relation may be viewed as incorporating the canonical commutation relations on a Cauchy surface, together with the equation of motion 1( .1).) In other words, our algebra A(M, g) consists of polynomials in the tf;(F) where elements (and a.djoints of elements) are declared equal if they can be reduced to one another by application of the rules of* algebras together with the rules (i)-i( v) (5). The elements of A(M,g) thus correspond to sums of suitably smeared products of fields at dif f erent points,

and specification of a state s( ee Note (l})w on A(M, g) amounts to specifying all its smeared n-point functions w(tfo(F1) ,P(Fn)). . • •

QFT ON (NON (lLOBALLY HYPERBOLIC) SPACETIMES

179

This minimal algebra really depends on our choice c of time orientation, and when we wish to make this dependence explicit, we shall write Ac(M,g). Had we chosen the opposite orientation -c, the algebra A_c(M·, g) we would have obtained would not be isomorphic to Ac(M,g). More precisely, one sees from (2.1) and (3.1) that the map tjJ(F) 1-+ tjJ(F), when regarded as a map between Ae(M,g) and A-e(M,g), fails to extend to an isomorphism, but rather extends to an anti-linear isomorphism where we say that a map T between two* algebras is an anti-linear isomorphism if it is bijective and if

=

I

r(-IA + JtB)

=

h(A)+ pr(B)

(C)

r(A•)

=

(r(A))'

(D)

r(AB)

=

(J\)

r(J)

(B)

r(A)r(B)

for all elements A, B of the first algebra and complex numbers .\ and Jl.. A(M,g) may be given the structure of a net of local* algebras by defining, for each open.region with compact closure 0 C M, the subalgebra A(M,g;O) gene­ rated by the tjJ(F) for F supported in 0 and, as we mentioned in the introduction, these will satisfy a. number of general properties including (see also [12]) (a) (isotony) 0 c CY o> A(M,g;O) c A(M,g; 0') (b) (causality) 0 in the domain of determinacy (see the definition in Note {4))

'

of 0' o> A(M,g;O) C A(M,g;O ) (c) (commutativity at spacelike separation) 0 and 0' spacelike separated::::}

[A(M,g;O),A(M,g;O')]

=

0.

Here, we point out the further property ofF-locality. To motivate this notion, we first focus attention on the following question: Let (N,g) be a globally hyperbolic subspacetime of (M,g) (We shall assume (N,g) inherits the time orientation from the choice of time-orientation on (M, g)). Then one can conceive of two ways of constructing a minimal algebra for equation (1.1) on N:

Either (1) One can restrict A(M,g) to the subalgebra A(.M,g;N) generated by rjJ(F) for F supported in N, or

(2) One may construct the algebra A(N,g)) by following the above procedure, viewing (N,g) as a globally hyperbolic spacetime in its own right. Question. What is the relationship between these two procedures?

It turns out that, in general one does not obtain equivalent algebras: For example, one could take (M,g) to be the timelike cylinder spa.::etime and (N,g) to be the helical strip subspacetime (see Fig. 3) described in Sec. 2. One will clearly get inequivalent algebras in this case since the restriction of the advanced and retarded

B.S.KAY

180

(M,g) to (N, g) will clearly not,

Greens functions from

in this case, coincide with

the intrinsic advanced and retarded Greens functions for However, it follows

from

same algebra provided that

the theorem in Sec.

N is

2

(N,g).

that one always

does obtain the

"sufficiently small" and furthermore this holds for

(M,g) is the (N,g) to be a double cone region

"arbitrarily small" neighbourhoods. For example, in the case where timelike cylinder, one can take, around any point, as sketched in Figlll'e 1

(6).

We call this the F-locality property.

The F-Locality Property (for real linear scalar fields on globally hy­ perbolic spaeetimes). Every neighbourhood of every point p in M contains a globally hyperbolic subneighbourhood N of p such that the subalgebra A(M, g;.N) generated by

{F)

for

F

with support in C8"(N) is equal (under identification of

elements represented by the same polynomial) to

A(N,g).

4. Quantum Field Theory on Globally Hyperbolic Spacetimes:

Second Level of Structure At first sight, it might appear reasonab� to take the conceptual and mathe­

(1.1) on a globally hyperbolic A(M, g) together with the linear functionals) won A(M,g).

matical framework for the quantum field theory of

spacetime as consisting simply of the minimal* algebra set of all states (i.e. positive

However, this would be unsatisfactory (as indeed it would be unsatisfactory even in the special case of flat spacetime!) for two (related) reasons: (1) The minimal* algebra

A(M,g) is

but nothing that could correspond to products of fields at the same point to a quantum energy momentum tensor

(2)

The set of all states on

A(M,g)

3, it rP(z)t/J(y) rP(z)2 (or

too small. AB we discussed in Sec.

contains (smeared versions of) products of fields at dife f rent points such as

T14v(x)). is too large. It contains e.g. states with

"infinite energy density". A clue as to what is needed is given by returning to the special case of our equation (1.1) in flat spacetime. In this case, one has a preferred ground state and its corresponding Fock representation and one could e.g. obtain a preferred class of physically admissible states by considering say all multiparticle states built up in the usual way by acting on the Fock vacuum with creation operators smeared with smooth one-particle wave functions. One feature that all the resulting states will have in common is the short distance behaviour of their "unsmeared" n-point functions.

For example, in the case of the two point function, one knows that

for all these physically admissible states the two point function takes the form:

w(t/J(x)t/J(y)):::; -(1/2·nV(l/o-+vlno-+w) where o- denotes the square of the space­ time interval between x and y and v and ware smooth two point functions, of which v depends only on the mass of the field and not on the state, while all the infor­ mation about the state is in the smooth piece behaviour for a wide

class of multiparticle

w.

More generally the full singular

and other physically admissible states in

fiat spacetime is the same as that of the vacuum state in the sense that the trun� cated (see Note

(8)) n�point functions for n other than two

will be smooth while the

181

QFT ON (NON GLOBALLY HYPERBOLIC) SPACETIMES

truncated two point function will have the same singula:rities as the vacuum state. It is this universal short distance behaviour that permits one to define quantities such as a normal ordered (M) and irnpose the relations (see Sec. 3) {i) (Hermiticity), (ii) (linearity) and (iii) (satisfaction of the field equation in a suitable weak sense). However, there will be no obvious replacement for the advanced minus retarded fundamental solution .6. and yet, presumably, some extra relations are needed to replace the commutation relations (iv) of Sec. 3. ·we shall approach this problem by demanding that the laws in the small of A(M, g) should coincide with the local laws obeyed by the minimal algebra in the globally hy11erbolic case. One possible way to make this notion precise would be to elevate the F-locality property isolated in Sec. 3 for the globally hyperbolic case to an F-locality condition: The F·locality Condition (for real linear s�::alar fields on non globally hyperbolic spacetimes). Assume given a * algebra A(M,g) whose elements are polynomials in abstract objects q)(F), as F ranges over Cg"(M), and which satisfies the relations (i), (ii) and (iii) of Sec. 3. We say A(M,g) satisfies F-locality (for equation (1.1)) if each point p in M has a globally hyperbolic neighbourhood (N,g) for which the subalgebra A(M, g;N) of A(M, g) generated by �(F) for F with support in ego (N) is equal (under identification of elements represented by the same polynomial) to Ac(N, g) (as constructed in Sec. 3) for some choice, c, of time-orientation on N.

We remark: (a) By combining this s1.atement with the F-locality property for globally hyper­ bolic space-tim.e'3, one may conclude that each neighbourhood of any point p in M contains a globally hyperbolic subneighbourhood (N,g) of p for which the subal­ gebra A(M,g; N) of A( M, g) generated by tfo(F) for F with support in N is equal (under identification of elements represented by the same polynomial) to Ac(N,g) for some choice, c, of orientation on N.) (b) We have allowed in this definition for the possibility of independent choices of time orientation on each subspacetime in order not to exclude a priori the possi­ bility of quantizing on non time orientable spacetimes. (The related, but different, question of space orientability is discussed i.n Note (3).) However we shall see below that this was in vain! (c) One easily sees that it is actually unnecessary 1;o assume a priori that A(M, g) satisfies relation (iii) of Sec. 3 (that the smeared Helds satisfy the field equation

188

B. S. KAY

weakly) since this will follow anyway from the resulting definition of F-locality. (d) Clearly, F-locality determines the cormnutator of two smeared fields lfo(F1), tjJ (F2) to be i times the intrinsic advanced minus retarded fundamental solution whenever the supports of F1 and F2 both lie in sufficiently small globally hyperbolic neighbourhoods of each point. However, there is no control on the size of these neighbourhoods. We shall say that a spacetime (M, g) is F-quantum compatible (for our Eq. (1.1)) if it admits an algebra A(M, g) which satisfies the F-locality condition. AB explained in the introduction, one could argue that a spaeetime which failed to be F-qua.ntum compatible in this sense could not arise as the classical limit of some state of quan­ tum gravity and hence should not occur in nature. However, as was also emphasized in the introduction, it may well be that F-locality should be replaced by a. weaker condition in which case one would have to modify the F-quantum compatibility condition accordingly. We emphasize, also, that F-locality should really only be regarded as a (pos­ sible candidate for a) necessary condition for quantum compatibility. To clarifY this point, we give a (two dimensional) example of a globally hyperbolic spacetime (M,g) and a set of relations on smeared quantum fields such that the resulting * algebra is consistent with F-locality, but is not globally equivalent to the usual minimal algebra: Take (M, g) to be a two dimensional double cone region, but re­ gard it as a. subspacetime of the timelike cylinder spacetime (M', g) by identifying it isometrically with a helical strip region (see Fig. 3 and the discussion in Secs. 2 and 3) and take, instead of the usual minimal algebra for (M, g), the restriction to (M,g) of the usual minimal algebra for (M',g). It is easy to see that this * algebra has the claimed properties. Presumably, this is the "wrong'' minimal algebra for the double cone. It could be ruled out by demanding some extra conditions - for example commutativity at spacelike separation. (Note, however, that, in the case, for example of the spacelike cylinder, the condition of commutativity at spacelike separation would be empty of content, since, in this spacetime, no two points are spacelike separated.) However, it is less clear, in general what constitutes a full set of sufficient conditions for quantum compatibility. One may well have to ap­ pend conditions which refer to the second level of structure {cf. Sec. 4). (One can even speculate that a reasonable such full set would, in the end force all quantum compatible spacetimes to be globally hyperbolic after. all!) Next, we list a numbex of :first consequences of these definitions. (1) Any globally hyperbolic spacetime is F-quantum compatible. This is the content of the F-locality property of Sec. 3. (2) Furthermore, any subspacetime of a globally hyperbolic spacetime is F­ quantum compatible. A (non globally hyperbolic) example is the timelike strip spacetime. However, there is a difference between the globally hyperbolic case and, say, the case of the timelike strip. In the globally hyperbolic case,· there are only two ways to quantize - i.e. two possible algebras1 one for each choice of time ori­ entation (see however the example in the previous paragraph). On subspacetimes such as our timelike strip, there will be a lot more non-uniqueness. E.g. one could

QFT ON (NON GLOBALLY HYPERBOLIC) SPACETIMES

189

embed the timelike strip in other globally hyperbolic spacetimes. For instance, one could modify Minkowski space in ,a compact region outside of the timelike strip to have non-zero curvatmre there; but in such a way that the spacetime remains globally hyperbolic. This wOuld change the subalgebra (say for a particular choice of time orientation) induced on the timelike strip. This is an easy consequence of the following classical consideration: There will exftst a choice of Cauchy surface C in the modified Minkowsl

0

will - under the conformal isometry between this region

and the full timelike cylinder (see Sec.

2)

- include a "semi-infinite" helical strip on

the timelike cylinder, similar to that drawn in Fig.

3, but infinitely extended in one

direction. Next, assume there is some algebra A(Misner) satisfying conditions (i), (ii) and (iii) of Sec. 3. One can use the conformal isometry to identify the restriction of this field algebra on Misner to the region

t

>

0 with a field algebra A(cyl) on (ii), and

the timelike cylinder whose smeared fields will also satisfy conditions (i),

(iii) . In particular, by using the conformal invariance of the two dimensional wave equation (see Note

(2)),

the smeared fields in A(cyl) on the timelike cylinder will

also satisfy the two dimensionaL wave equation on the timelike cylinder weakly. Moreover, the assumption ofF-locality of A(Misner) at points with

t

>

0 will imply

F-locality of A(cyl) on the timelike cylinder while the assumption ofF-locality at a point p on the surface

t =0

in Misner will imply, by our initial remark, that

QFT ON (NON GLOBALLY HYPERBOLIC) SPACETIMES

there is some semi-infinite helical strip H in the timelike cylinder on which the restriction of A(cyl) coincides with the intrinsic A(H,g) constructed as in Sec. 3. One can now get a contradiction by exhibiting three test functions F1, F2 and F� on the timelike cylinder with supports in H such that (a) F2 and F2 are related by F2 - Fi DG where G is another test function, so that, using the fact that �(OG) O,�(F2) �(FD (b) by u'ing the '"!." of A(H,g), the commutator [I,O(F!), 1,6(F2)] vanishes, while, the assumption of F-locality about another point p in the timelike cylinder implies that the commutator [Ql(Ft), qi(F:j)] does not vanish. (a) and (b) can be achieved by first choosing the point p in H and choosing F1 and F:2 to both be supported in a globally hyperbolic subneighbourhood on which one knows, by F-locality, that their conunutator doesn't vanish, and then choosing F2 to the past of F:2 in the timelike cylinder, but in such a way that they are related as in (a) and their supports are spacelike separated in the intrinsic geometry of H. It is not difficult to show that this can be done by using a construction similar to that in "Note 1" in [17] and relying on the special features (propagation on the light cone of the derivative of a solution) of the two dimensional wave equation (on the timelike cylinder). We remark, concerning the argument in (3) for the non F-quantum compatibility of non time orientable spacetimes, that is seems likely that this result might survive even if one considerably weakened the notion of F-locality, as we discussed in the introduction. Concerning the last two examples (F-quantum compatibility for the four dimen­ sional spacelike cylinder, and non F-quantum compatibility for two dimensional Misner space - both in the case of a massless field theory), obvious questions are whether the Misner space result extends to the four dimensional case (where one products either with two dimensional Euclidean space or a two dimensional space­ like torus) and whether these results would still hold in the case of massive field theories. The argument for F-quantum compatibility of the four dimensional space­ like cylinder given above would clearly break down in the massive case because of propagation "inside the light cone". It is an open question whether F-quantum compatibility holds in some other way in this case. One possibility is that strict F-quantum compatibility might fail, but that some other weaker notion may hold. On the other hand, it seems plausible that the argument for the non F-quantum compatibility of Misner space given above could be modified to deal with the mas­ sive case (or four dimensions) and may also survive some weakening of the notion of F-locality. The Misner space example is interesting in connection with the recently discussed question as to whether it is possible in principle, when one takes into account quantum effects, for spacetime to "develop" regions with closed timeiike curves (see e.g. [22], [23J and references therein). In [2 2] Hawking isolates a general class of such spacetimes with an ''initial" globally hyperbolic region and with a "compactly generated Cauchy horizon" and his Chronology Protection Conjecture [22] is that such spacetimes cannot be approximate classical de.ocriptions of states of quantum gravity. The argument in favour of this conjecture given in [22] involves looking at =

=

=

192

B. S. KAY

the expectation value for the renormalized energy momentum tensor (see Sec.

4)

for states in the "initial" globally hyperbolic region. As discussed in [22] , in many cases which have been studied, this will diverge, in such a way as to give a large back-:reaction on the geometry, on the Cauchy horizon. From the standpoint of a semidassica.l description of these spacetimes in terms of quantum field theory in curved spacetime, it is thus argued that something goes wrong with these spacetimes at the level of physically admissible states (what we have called here the "second level of structure"). If it turns out that our result of non F-quantum compatibility (or maybe, as we have discussed, one would prefer some weaker notion) extends (i.e. from the two dimensional massless Misner space example treated here) to other spacetimes in the class discussed in [22] it would indicate that something goes wrong with these spacetimes already at the "first level of structure". Notes (1} By a * algebra A we shall always intend an algebra with identity I over the complex numbers with a * operation satisfying

A"'"' = A, (AB)* = B* A*

and

for all complex numbers >.. , /' and elements of the

(.\A + p.B)"' = (XA* + fjB"')

algebra A, B. By a net of * algebras we shall mean a collection of subalgebras (i.e. * subalgebras with identity) A(CJ) of some * algebra ·indexed by the set of open

regions with compact closure

0 of some spacetime manifold and satisfying isotony 0 C 01 ==? A(O) C 'A(O') possibly together with Sl)me other relations. In Secs. 3 and 6, we shall require nothing more from the theory of * algebras than these definitions, while for Secs. 4 and 5 we shall assume familiarity with little more

i.e.

than the following elementary notions: (a) The definition of a state

0,\fA E A)

w

on a * algebra .A as a positive (i.e.

w(A'"A) ;?:

linear functional.

(b) The notion of

pure and mixed states: A state w is said to be mixed if it can w = aw1 + (1 - a)w2 (0 < a < 1) of other

be expressed as a convex combination states, and pure otherwise. (c) The

This associates to any state w a triple

GNS construction.

a representation on a common dense domain in the Hilbert space vector the

*

Ow)

such that

w(A) = (flw1Pw(A)flw)1tw ·

(Pw, 1f.w, fiw)(Pw 1iw with cyclic

(Roughly speaking - it is true if

algebra is a C" algebra - the GNS representation of a state is irreducible if

a.nd only if the state is pure.) (d) Following Fredenhagen and Haag [2}, we define the consist of the set {w6 : 8

>

0, tr8

= l,w5(A) =

folium of a

state

w to

tr(Sp141(A))?t:..,} of density matrix

states in the GNS representation belong:4tg to w. We refer to [1] for more discussion and for discussion of a few other topics (e.g.

c• and

W"'

algebras and factors and the ground state and KMS conditions for

automorphism groups and the notion of quasi-equivalence) we shall mention in passing in Secs.

4 and 5.

(2} A conformal isometry between two spacetimes

(M1,g1), (M2 , g2) is a diffeo-­ 5.,g1 induced by 2 the diffeomor:phism is conforrnally related to the existing metric g2 - i.e. 8*g1 0 g2 mo:rphism 0

:

M1

-r

M2

with the property that, on M2 the metric

=

QFT ON (NON GLOBALLY HYPERBOLIC) St>ACETIMES

193

for some smooth strictly positive function n on M2 • We remark also that, in general, conformally related metrics have the same causal structures and, in two-dimensions, the Laplace Beltrami operators (and hence the massless scalar field equation) for conformally related metrics are identical. (3) The volume element dTJ as defined in Sec. 2 makes sense whether or not the spa.cetime is orientable. In particular, while globally hyperbolic spacetimes are necessarily time-orientable, we do not assume they are necessarily space-orientable (see e.g. [11]), and it appears that the advanced and retarded Green's functions, and hence the construction ofthe minimal algebra in Sec. 3, do not require such an assumption either. An alternative, more cautious approach to non space-orientable globally hyperbolic spacetimes, would be to accept the minimal algebra construction of Sec. 3 at first only for the space-orientable globally hyperbolic spacetimes (note that the algebras thus constructed for the two choices of space-orientation would be isomorphic) and to modify the F-locality condition of Sec. 6 accordingly. Clearly, in the case of non space-orientable globally hyperbolic spacetimes, the minimal algebra constructed as in Sec. 3 here would satisfy the resulting notion of F-locality. {4) To see, in general, that every point has a globally hyperbolic neighbourhood, one can work in geodesic normal coordinates for p aJCld take the domain of detenni­ nacy N of an open 3-ball of constant time in these coordinates which lies in a single convex normal neighbourhood and has sufficiently small radius so that N also lies in this neighbourhood. (The domain of determinacy of any set S in a time-orientable spacetime is defined to consist of the union of set of all points to the future of the S for which every past directed causal curve intersects S with the set of all points to the past of S for which every future directed causal curve intersects S.) {5) For some purposes, it is more convenient to take a slightly different definition for the minimal algebra. For example, one can take the Weyl algebra over the quotient of CO (M) by (09 + m2) CO(M) with the symplectic form o-([F1], [F,]) a{F11 F2) (here [F] denotes the equivalence class of F). Here the Weyl algebra over a. symplectic space (S, u) is defined to be the * algebra generated by abstract objects W(a) , a E S satisfying W(O) 1, W(a)' W(-a) and W(a1)W(a2) exp(-iu(a1, a2)/2)W(a1 + a2) and has the virtue of having a unique C"" .norm, so that it can be completed to a C" algebra. Formally one may then identity W([F]) with exp(-irfl(F)). See e.g. [7] for further discussion and references. {6) In fact, it is interesting to note, in the example given just before the statement of the F-locality property in Sec. 3, that, were one to (isometrically) embed the same double cone region in Minkowski space rather than the cylinder, one would again obtain the same algebra, and thus it is in fact impossible in principle in this case, by making measurements in N, to distinguish between these two situations. This observation (which is reminiscent of the equivalence principle) was :first made in [13, 14] and applied there to a discussion of the theoretical basis of the Casimir effect � i.e. to the problem of fixing the renormalization ambiguity {see the discussion in Sec. 4 here) for the expecta.tion value of the energy momentum tensor in the cylinder ground state (see Note {7)). In particular, we pointed out there that one could not distingUish between the situation where such a double cone region is embedded in =

=

o=

=

B. S. KAY

194

the cylinder spacet.ime, with the state ofthe quantum field the cylinder ground state) and the situation wh·ere the double cone region is actually embedded in Minkowski space, but with the state of the field some quantum state in Minkowski space (not equal to the usual ground state in Minkowski space!) whose restriction to the double cone region coincides with the restriction of the cylinder vacuum state. With this point of view, one can then fix the renormalization ambiguity at any point in the cylinder, by regarding it as a point in such a double cone region, and in such a state, in Minkowski space and by normal ordering with respect to the Minkowski ground state.

(7} The notion of ground state does generalize to the class of globally stationary [21], i.e. to spacetimes which admit an isometry group whose integral.

spacetimes

curves are timelike (i.e. which admit an everywhere timelike Killing vector field) . w

{8) In this note, we recall the definition of the· "truncated n-point functions" (J(F1) . . . J(Fn))T of a state w (on some algebra of smeared quantum fields) and

then define quasifree states. To define the truncated n point functions, one can proceed inductively: First, one defines the truncated 1-point function to equal the ordinary 1-point funC­ Then, for any

tion.

n,

one assumes one has defined the notion for all

m