## Abstract

A generalized equivalence principle is put forward according to which *space-time symmetries and internal quantum symmetries are indistinguishable before symmetry breaking*. Based on this principle, a higher-dimensional extension of Minkowski space is proposed and its properties examined. In this scheme the structure of space-time is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time (QST) possesses a rich hierarchical structure. The natural extension of the Poincaré group to QST is investigated. In particular, we prove that the symmetry group of this space is generated in general by a system of irreducible Killing tensors. After the symmetries are broken, the points of the QST can be interpreted as *space-time valued operators*. The generic point of a QST in the broken symmetry phase then becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from QST to Minkowski space. It is shown that the general such map satisfying appropriate causality-preserving conditions ensuring linearity and Poincaré invariance is necessarily a density matrix.

## 1. Introduction

The purpose of this paper is to present a novel approach to the unification of space-time physics and quantum theory. We take the view that classical space-time itself is not to be regarded as a primary object that is subjected to some form of quantization procedure. In our framework, the central object is a mathematical structure called a quantum space-time (QST). Intuitively, this structure can be regarded as the space of all space-time valued quantum operators. That is, each point in the infinite-dimensional QST corresponds to a quantum operator with the property that its expectation, in any quantum state, is a space-time point. The space of all such operators has a rich structure that appears to contain all the elements one needs for both a characterization of the causal structure of relativistic space-time as well as a representation of the phenomena of quantum theory.

Many attempts to unify gravitational physics with other fundamental forces have pursued the idea of extending four-dimensional space-time to higher dimensions. Beginning with the introduction of the Kaluza–Klein theory of gauge potentials, such extensions have typically been carried out by increasing the spatial dimension of the space-time, while retaining the special role played by time. In methodologies of this sort it cannot be said that quantum mechanical characteristics of the fundamental forces are adequately incorporated into the structure of the higher dimensional space-time. Nor can it be said that the space-time is being treated in any useful sense as a quantum entity. In what follows, however, we demonstrate that if the universe is intrinsically quantum mechanical in an appropriate sense, then the most natural extension of space-time into higher dimensions has a completely different character from that suggested by the Kaluza–Klein theory and its generalizations.

The basic idea is as follows. The points of Minkowski space are in natural correspondence with two-by-two matrices of the form *x*^{AA′} satisfying a Hermitian condition. Lorentz transformations are then given by multiplying *x*^{AA′} on the right and on the left by an element of and its complex conjugate, and the Minkowskian metric for the interval between two points *x*^{AA′} and *y*^{AA′} is obtained by taking the determinant of their difference. Hermitian matrices are, on the other hand, familiar objects in quantum mechanics in their role as physical observables. In quantum mechanics, the dimensionality of the Hilbert space is directly related to the dimensionality of the corresponding space of Hermitian operators. Thus, if quantum theory is to be unified with space-time physics, it is natural to extend the space of matrices representing space-time points to higher dimensions, and hence to assume that the dimensionality of space-time is larger than four, perhaps infinite. A higher-dimensional extension of space-time in this manner is also consistent with the philosophy often put forward that spinors are just as fundamental as space-time points.

The framework we introduce here is motivated by the idea that the symmetries of space-time and the symmetries of quantum theory are, at the deeper level, indistinguishable. This generalized ‘equivalence principle’ reflects the notion that the fundamental symmetries or approximate symmetries we observe in nature should have a common origin, and that the breakdown of these symmetries should also have a common cause. In view of the generalized equivalence principle, we therefore postulate that *space-time events are themselves infinite-dimensional Hermitian matrices*. As in ordinary quantum mechanics, however, it is desirable to consider finite-dimensional realizations of the framework in some circumstances. These finite-dimensional realizations are given by spaces of *r*-by-*r* Hermitian matrices. In this way we obtain for each *r*≥2 an *r*^{2}-dimensional QST which we denote . The standard four-dimensional Minkowski space ^{4} then emerges as the simplest case.

As we explain later in the paper, we regard such finite-dimensional cases not merely as toy models, but rather as special situations where a finite-dimensional part of the infinite-dimensional space-time is (or effectively can be regarded as) disentangled from the rest of the space-time. In this way, the fundamental role played by the Segré embedding in the geometry of quantum theory is carried over to the relativistic domain. Indeed, it is an important feature of our approach that many of the familiar ideas relevant to the geometry of the quantum state space are applicable to space-time itself, and as such, take on a new physical significance, some aspects of which we explore here.

The structure of the paper is as follows. In §2 we introduce the algebraic formalism appropriate for *r*-component hyperspinors. The concept of hyperspinors as a higher-dimensional generalization of the familiar two-component spinors of relativity theory and as a basis for higher-dimensional space-time theories was introduced by Finkelstein (1986) and has subsequently been the subject of extensive investigation (Finkelstein *et al*. 1986, 1987; Holm 1986, 1988*a*,*b*, 1990; Finkelstein 1988; Borowiec 1989, 1993; Urbantke 1989). The space , the main object under consideration in this paper, arises as the tensor product of the space of *r*-dimensional hyperspinors with its complex conjugate. In §3 we investigate the causal structure of . This structure is shown to arise by virtue of a resolution of the vector separating any two points in into a canonical form involving a sum of terms, each of which can be expressed as a product of a hyperspinor with its complex conjugate, together with a plus or minus sign. In particular, we introduce the concept of future and past pointing time-like and null vectors in . This structure exists even though does not possess a pseudo-Riemannian metric. Instead, possesses a chronometric form of rank *r*, which induces a pseudo-Finslerian geometry on . In §4 we look at the variational problem for determining the geodesic between two time-like separated points, and show that this reduces to an appropriate linear expression, despite the fact that the chronometric form is itself a polynomial of degree *r* in the separation vector for the given points.

In §§5 and 6 we study the higher dimensional analogue of the Poincaré group that acts on . We prove that the symmetries of this space are generated by a system of 3*r*^{2}−2 irreducible Killing tensors, each of rank *r*−1. Since does not have a pseudo-Riemannian structure for *r*>2, the link between symmetries and Killing vectors is replaced by this more subtle manifestation of symmetry. We show that the conserved quantities associated with the hyper-Poincaré group can be obtained in terms of algebraic expressions formed from the Killing tensors. We also derive appropriate hyper-relativistic generalizations of the familiar expressions for the momentum, angular momentum, mass and spin of a relativistic system.

In §7 we study the algebraic geometry of the complex light-cone at a point of , and examine the structures arising for various values of *r*. We show that the space of complex light-like directions is a complex hypersurface of degree *r* in the complex projective space . The hypersurface can be completely characterized by the fact that it admits a special hyperspinorial subvariety of the form . In §8 we present a higher dimensional analogue of the Klein representation, and show how , when complexified and compactified, can be represented as the Grassmannian of complex (*r*−1)-planes in . We show that the causal relations between points of can be understood in terms of the intersection properties of the corresponding (*r*−1)-planes.

In §9 we show how the conformal symmetry of the geometry of the generalized Klein representation can be reduced to the hyper-Poincaré group by the introduction of elements determining the structure of infinity for this space. The range of possibilities for structure at infinity is considerably larger than it is for four-dimensional space-time. As a consequence, as we show in §10, the choice of structure at infinity can also give rise to interesting classes of cosmological models. We point out that there is a mechanism within our framework whereby the same structure at infinity responsible for the reduction of the symmetry of space-time is responsible for the breaking of microscopic symmetries. (Hyper-cosmologies have also been investigated, from a slightly different point of view, by Finkelstein *et al*. (1987) and Holm (1988*a*,*b*)).

In §11 we further explore the notion of symmetry breaking, and introduce the idea of the Segré embedding as the basis of the mechanism according to which space-time degrees of freedom can be disentangled from the microscopic or internal degrees of freedom. According to this scheme, the dimension of the hyperspinor space is assumed to be even, and each hyperspinor index *A* with the range *A*=1, 2, …, 2*n* is regarded as a clump consisting of a conventional two-component spinor index * A*=1, 2 and an ‘internal’ index

*i*=1, 2, …,

*n*. It follows as a consequence of this symmetry-breaking scheme that the points of can be interpreted as

*space-time valued operators*. This is the sense in which can be regarded as a QST. Finally, in support of this interpretation, in §12 we consider maps from to Minkowski space

^{4}. We show that if such a map

*ρ*is in a suitably defined sense (i) linear, (ii) Poincaré invariant, and (iii) causal, then

*ρ*is a density matrix, and the map can be interpreted as the quantum expectation. Thus, in our framework, an important element of the probabilistic interpretation of quantum theory arises as an emergent property deriving from the causal nature of space-time.

## 2. Hyperspinors

What we aim for here is not a higher-dimensional pseudo-Riemannian analogue of four-dimensional space-time, but rather a geometry of a different character that, although richer in structure than Minkowski space, nevertheless retains a definite relation to that space—and as a consequence, is in a position to embrace the description of physical phenomena, albeit in a new setting. We find it convenient to proceed in stages. The first step is to introduce a special type of causal geometry of dimension *r*^{2}, where *r*≥2 is an integer. Later, we specialize to the case for which *r* is even, and introduce some additional structure that cements the relationship of the higher-dimensional space to Minkowski space. We demonstrate that when *r* is even there exists a natural symmetry-breaking mechanism that embeds Minkowski space in the higher-dimensional space. By virtue of the geometry of this embedding we can assign physical properties to elements of the higher-dimensional space.

For general *r*, we refer to this space as , the QST of dimension *r*^{2}. If we fix a point of origin in , then a point of can be characterized by its position vector with respect to that origin. In Minkowski space ^{4} such vectors are naturally isomorphic to elements of the vector space obtained by taking the tensor product of a complex vector space with its complex conjugate . We recognize and as the spaces of unprimed and primed two-component spinors, respectively. For two-component spinors we use bold roman indices, and we adopt the usual conventions for raising and lowering indices, and for complex conjugation. Thus if , then we write and , where . Likewise, for the complex conjugation map we write , where . A special feature of the two-component spinor algebra is that the epsilon spinor functions as a non-degenerate symplectic form that can be used to establish a linear map from the spin space to its dual space . For further details of the two-component spinor algebra, see, for example, Pirani (1965), Penrose (1968) and Penrose & Rindler (1984/1986).

Our model of QST generalizes the two-component spinor formalism and its relation to Minkowski space by allowing the underlying spin spaces to be higher-dimensional complex vector spaces, the elements of which, using the terminology of Finkelstein (1986), we call *hyperspinors*. The elusive notion of unifying quantum theory with the space-time concept into an intrinsically quantum-mechanical object—QST, as we call it here—has been pursued by many authors from various points of views (e.g. Finkelstein 1996 and references therein). Indeed, this idea was evidently a key motivation for the introduction of hyperspinors (cf. Finkelstein 1986). Let us write and , respectively, for the complex *r*-dimensional vector spaces of unprimed and primed hyperspinors. For hyperspinors we use unprimed and primed italic indices. It is assumed that these spaces admit an anti-linear isomorphism under the operation of complex conjugation. Thus, if , then under complex conjugation we have , where . In a standard basis this map conjugates *α*^{A} component by component to give the components of .

The dual spaces associated with the hyperspin spaces and will be denoted as and , respectively. If and , then their inner product is *α*^{A}*β*_{A}. Likewise, if and , then for their inner product we write *γ*^{A′}*δ*_{A′}.

We also introduce the totally antisymmetric hyperspinors of rank *r* associated with the spaces , , and . These will be denoted *ϵ*^{AB…C}, *ϵ*_{AB…C}, *ϵ*^{A′B′…C′}, and *ϵ*_{A′B′…C′}. The choice of these antisymmetric hyperspinors is canonical up to an overall scale factor. Once a specific choice has been made for *ϵ*_{AB…C}, then the other epsilon hyperspinors are determined by the relations , , and , where is the complex conjugate of *ϵ*_{AB…C}. If we introduce a standard basis then it is convenient to set *ϵ*_{12…r}=1, which is sufficient to fix the remaining components of the epsilon hyperspinors. The arguments that follow do not depend on a specific choice of scale.

The epsilon hyperspinors play a role similar to that of the two-index epsilon spinors of the two-component spinor algebra; but it should be evident that in the case of *r*-component hyperspinors the algebra is more elaborate. In particular, for *r*≥3 the epsilon spinor no longer has an interpretation as a symplectic structure.

Next, we introduce the complex matrix space . An element is *real* if it satisfies the weak Hermitian property , where is the complex conjugate of . We denote the linear space of real elements of by . The elements of constitute the real QST of dimension *r*^{2}. We regard as the complexification of .

## 3. Chronometric relations on

Consider two points *x*^{AA′} and *y*^{AA′} in , and write *r*^{AA′}=*x*^{AA′}−*y*^{AA′} for the corresponding separation vector, which is independent of the choice of origin. In what follows, we find it useful to introduce an index-clumping convention (Penrose 1968; Penrose & Rindler 1984), and write *a*=*AA*′, *b*=*BB*′, and so on, according to which a pair of hyperspinor indices, one primed and the other unprimed, corresponds to a lower case single vector index. Thus we set *x*^{a}=*x*^{AA′}, *y*^{a}=*y*^{AA′}, *r*^{a}=*r*^{AA′}, and so on. Then for the separation vector of the points *x*^{a} and *y*^{a} in we write *r*^{a}=*x*^{a}−*y*^{a}.

There is a natural causal structure induced on by the so-called *chronometric tensor g*_{ab…c} (Finkelstein 1986; Finkelstein *et al*. 1986). Making use of the index-clumping convention, we define this tensor as follows:(3.1)The chronometric tensor, which is of rank *r*, is totally symmetric and is non-degenerate in the sense that for any vector *r*^{a}, the condition *r*^{a}*g*_{ab…c}=0 implies *r*^{a}=0. We shall say that *x*^{a} and *y*^{a} in are *null separated* if the *chronometric form* for their separation vanishes: . Null separation is equivalent to the vanishing of the determinant of the matrix *r*^{AA′}:(3.2)

If the hyperspin space has dimension *r*=2, this reduces to the usual condition for *x*^{a} and *y*^{a} to be null-separated in Minkowski space. For *r*>2, however, the situation is more complicated on account of the fact that there are various degrees of nullness that can prevail between two points. More precisely, when two points of QST are null-separated, we define the ‘degree’ of nullness by the rank of the matrix *r*^{AA′}. Null separation of the first degree is the case for which *r*^{AA′} is of rank one, and thus satisfies a system of quadratic relations of the form(3.3)or equivalently, *g*_{ab…c}*r*^{a}*r*^{b}=0. This implies in the case of a real separation vector that *r*^{AA′} can be expressed in the form(3.4)for some hyperspinor *α*^{A}. If two points have a separation vector of this form then we say that they are *strongly* null separated. If the sign is positive (*resp*. negative), then *x*^{a} lies to the future (*resp*. past) of *y*^{a}.

In the case of nullness of the second degree, *r*^{AA′} satisfies a set of cubic relations given by *g*_{abc…d}*r*^{a}*r*^{b}*r*^{c}=0. In this case *r*^{AA′} can be put into one of the following three forms: (i) , (ii) , and (iii) . In case (i), *x*^{a} lies to the future of *y*^{a}, and *r*^{a} can be thought of as a ‘degenerate’ future-pointing time-like vector. In case (ii), *r*^{a} can be thought of as a degenerate space-like separation. In case (iii), *x*^{a} lies to the past of *y*^{a} and *r*^{a} is a degenerate past-pointing time-like vector. A similar analysis can be applied in the case of null separation of other ‘intermediate’ degrees.

If the determinant of *r*^{AA′} is non-vanishing, and *r*^{AA′} is thus of maximal rank, then the chronometric form is non-vanishing. In that case *r*^{AA′} can be represented in the canonical form(3.5)with the presence of *r* non-vanishing terms, where the *r* hyperspinors *α*^{A}, *β*^{A},…,γ^{A} are linearly independent.

Let us write (*p*,*q*) for the numbers of plus and minus signs appearing in the canonical form for *r*^{AA′} given in equation (3.5). We shall call (*p*,*q*) the signature of the vector *r*^{AA′}. The hyperspinors *α*^{A}, *β*^{A},…,γ^{A} are determined by *r*^{AA′} only up to an overall unitary (or pseudo-unitary) transformation of the form , where *n*,*m*=1, 2,…,*r* and . The signature (*p*,*q*) is, however, an invariant of *r*^{AA′}.

In the cases for which *r*^{AA′} has signature (*r*,0) or (0,*r*), we say that *r*^{AA′} is time-like future-pointing or time-like past-pointing, respectively. Then writing(3.6)for the associated chronometric form, we define the time-interval between the events *x*^{a} and *y*^{a} by the formula . It should be evident that in the case *r*=2 we recover the standard Minkowskian time-interval between the given events.

In summary, we have the following classification scheme for the separation between two space-time points. Let *r*^{2} be the dimension of the QST and (*p*,*q*) the signature of the separation vector *r*^{a}. If *p*+*q*=*r* then we say that the separation is non-degenerate; then if *p*=*r*, the vector *r*^{a} is time-like future-pointing, and if *q*=*r*, then *r*^{a} is time-like past-pointing.

If neither *p* nor *q* equals *r*, then we say *r*^{a} is space-like of type (*p*,*q*). Note that in the case of ordinary Minkowski space, for which *r*=2, the fact that a space-like vector is necessarily of type (1,1) corresponds to the result that any space-like vector in four-dimensional space-time can be expressed as the difference between two real null vectors. Any two such representations for the same space-like vector are related by a *U*(1,1) transformation.

On the other hand, if *p*+*q*<*r*, then we say that *r*^{a} is a *degenerate future-pointing vector* if *q*=0, and a *degenerate past-pointing vector* if *p*=0, and otherwise a *degenerate space-like vector*. Clearly, all degenerate vectors are null in the sense that the corresponding chronometric form vanishes. If *p*=1 and *q*=0, then *r*^{a} is future-pointing and strongly null, and if *p*=0 and *q*=1 then *r*^{a} is past-pointing and strongly null. Strong null separation is the analogue of Minkowskian null separation. The measure of separation ‖*x*−*y*‖ is non-vanishing if and only if the separation vector is non-degenerate. The causal structure of the QST, however, also brings into play the various degenerate forms of time-like or null separation. Thus, if *x*^{a} lies to the future of *y*^{a} (i.e. if *x*^{a}−*y*^{a} is time-like future-pointing, degenerate future-pointing, or strongly null future-pointing), and if *y*^{a} lies to the future of *z*^{a}, then *x*^{a} lies to the future of *z*^{a}.

A striking feature of the causal structure of QST is that the essential physical features of the causal structure of Minkowski space are preserved. In particular, the space of future pointing time-like vectors forms a convex cone, and the convex hull of this cone includes the future pointing null vectors of all degrees of degeneracy.

## 4. Dynamical trajectories and geodesics

Now suppose that the map defines a smooth curve *Γ* in for . Then *Γ* will be said to be a time-like curve if the tangent vector(4.1)where the dot denotes differentiation with respect to *λ*, is time-like and future-pointing along *Γ*. Then, we define the proper time *s* elapsed along *Γ* by(4.2)For convenience, we can also write equation (4.2) in the infinitesimal form(4.3)This expression shows that the geometry under consideration here has a *pseudo-Finslerian* structure. Finslerian geometries, first considered by Riemann, and studied extensively by Finsler, have from time to time been proposed as the basis for generalizations of the theory of relativity. It is interesting that such a structure arises in a natural way in the present context. One should note, however, that the pseudo-Finslerian structures arising in our framework are of a very particular sort.

Let us now consider the properties of geodesics in a QST (see Holm 1986 and Borowiec 1993 for related work). For a time-like curve, we can choose the proper time as the parameter along the curve, in which case the resulting affine parametrization of the curve is determined up to a transformation of the form *s*→*s*+*c* where *c* is a constant. The equation of motion for the situation in which *Γ* is a time-like geodesic is obtained by varying equation (4.2) and setting the result to zero. Writing(4.4)we find that *x*^{a}(*λ*) describes a geodesic if the velocity vector *v*^{a} satisfies the Euler–Lagrange equation(4.5)A calculation shows that this condition is given more explicitly by(4.6)where *ϕ*=d ln *L*/d*λ*. If *λ* is chosen to be proper time, then *ϕ*=0 and the geodesic equation take the form(4.7)

In the case *r*=2, equation (4.7) reduces to the familiar relation . To prove that the geodesic equation (4.7) implies in the case *r*≥2, it suffices to examine the case *r*=3. Then we have , which can be expressed in hyperspinor terms as(4.8)This relation in turn can be written(4.9)However, because , we know that *v*^{AA′} has an inverse *u*_{AA′} satisfying and . Therefore, contracting equation (4.9) with *u*_{BB′} we obtain . This equation shows that if were not zero, then it would have to be proportional to . However, if that were so, then equation (4.8) would imply , contrary to the assumption that *v*^{CC′} is time-like. It follows that . That concludes the proof for *r*=3. A similar argument establishes for all *r*≥2 that equation (4.7) implies . Hence we have:

*Let A*^{a} *and B*^{a} *be quantum space-time points with the property that A*^{a}−*B*^{a} *is time-like and future-pointing*. *The affinely parametrized geodesic connecting these points in* *is the curve*(4.10)*for* −∞<*s*<∞, *where* .

## 5. The hyper-Poincaré group

The chronometric form *Δ* for the separation between two points in is invariant when the points of are subjected to transformations of the following type:(5.1)Here represents an arbitrary translation in QST, is an element of , and is its complex conjugate. The relation of this group of transformations to the Poincaré group in the case *r*=2 is evident. Indeed, one of the attractions of the extension of space-time geometry under consideration here is that the resulting hyper-Poincaré group admits such a description.

More generally, we observe that the (proper) hyper-Poincaré group preserves the signature of any space-time interval *r*^{AA′}, whether or not the interval is degenerate, and hence, leaves the causal relations between events unchanged.

We refer to a transformation of the form as a hyper-Lorentz transformation if for some element .

In the case of Minkowski space (*r*=2), it is well known that a two-component spinor *ξ*^{A} can be represented by a null bivector . Conversely, *Λ*^{ab} determines *ξ*^{A} up to the transformation *ξ*^{A}→−*ξ*^{A}. In the case of a QST of dimension *r*^{2}, a hyperspinor *ξ*^{A} can be represented by a null *r*-vector , and *Λ*^{ab…c} determines *ξ*^{A} up to transformations of the form , where *k*=1,2,…,*r*−1. (See Holm (1990) for further discussion of the hyper-Lorentz group, and the action on the group arising here.)

The (real) dimension of the hyper-Lorentz group is 2*r*^{2}−2, and the dimension of the hyper-Poincaré group is thus 3*r*^{2}−2. It is interesting to observe that the dimension of the hyper-Poincaré group grows linearly with the dimension of the QST itself, which is given by *r*^{2}. This can be contrasted with the dimension of the group arising if we endow with a standard Lorentzian metric with signature (1,*r*^{2}−1). In that case, the pseudo-orthogonal group has real dimension *r*^{2}(*r*^{2}−1), which together with the translation group gives a total dimension of *r*^{2}(*r*^{2}+1). The parsimonious dimensionality of the hyper-Poincaré group is due to the fact that it preserves the rather delicate system of causal relations holding between the points of QST.

## 6. Symmetries and conservation laws

In Minkowski space the symmetries of the Poincaré group are associated with a 10 parameter family of Killing vectors. That is, for *r*=2 we have the Minkowski metric , and the Poincaré group is generated by the 10 parameter family of vector fields *ξ*^{a} on ^{4} satisfying _{ξ}*g*_{ab}=0, where _{ξ} denotes the Lie derivative. Now for any vector field *ξ*^{a} and any symmetric tensor field *g*_{ab} we have(6.1)If *g*_{ab} is the metric and ∇_{a} denotes the associated Christoffel derivative satisfying ∇_{a}*g*_{bc}=0, we obtain , where *ξ*_{a}=*g*_{ab}*ξ*^{b}. The condition _{ξ}*g*_{ab}=0 therefore implies that *ξ*^{a} is a Killing vector.

We have taken the trouble to spell out the case *r*=2 in order to highlight the contrast with the situation for general *r*. For *r*>2 we have no Riemannian metric, and the usual relation between symmetries and Killing vectors is lost. What survives, however, is of considerable interest. More specifically, to generate a symmetry of , the vector field *ξ*^{a} has to satisfy , where *g*_{ab…c} is the chronometric tensor. Now for a general vector field *ξ*^{a} and a general symmetric tensor field *g*_{ab…c} we have(6.2)In the case of a QST we let ∇_{a} be the flat connection for which ∇_{a}*g*_{bc…d}=0. Then, to generate a symmetry of the chronometric structure the vector field *ξ*^{a} must satisfy(6.3)which serves as the analogue of Killing's equation. Equation (6.3) can be written in a more suggestive form if we define a symmetric tensor *ξ*_{ab…c} of rank *r*−1 by(6.4)Then equation (6.3) says that *ξ*_{ab…c} satisfies the conditions for a symmetric Killing tensor: . Thus we see that provides an example of a symmetry group generated by a family of Killing tensors.

*The symmetries of the quantum space-time* *are generated by a system of* 3*r*^{2}−2 *irreducible symmetric Killing tensors of rank* *r*−1.

The significance of Killing tensors is that they are associated with conserved quantities. A well-known example of a conserved quantity associated with an irreducible Killing tensor (that is, a Killing tensor that cannot be expressed as a sum of products of Killing vectors) is Carter's fourth integral of the equations of motion for geodesics and charged-particle orbits in the Kerr and Kerr–Newman solutions of Einstein's equations (Carter 1968; Walker & Penrose 1970; Hughston *et al*. 1971; Hughston & Sommers 1973; Penrose & Rindler 1986). In the present context it follows that if the vector field *v ^{a}* satisfies the geodesic equation, which on a chronometric space of dimension

*r*

^{2}is given by(6.5)and if

*ξ*

_{ab…c}is the Killing tensor of rank

*r*−1 given by equation (6.4), then we have the conservation law . In other words, is a constant of the motion.

Thus in higher-dimensional QSTs the apparatus of conservation laws and symmetry principles remains intact in the absence of a pseudo-Riemannian metric. In particular, the conservation of hyper-relativistic momentum and angular momentum for a system of interacting particles can be given a well-defined formulation, the basic principles of which are similar to those applicable in the Minkowskian case. For this purpose, we introduce the notion of an ‘elementary system’ or particle in hyper-relativistic mechanics. Such a system is defined by its hyper-relativistic momentum and angular momentum. The hyper-relativistic momentum of an elementary system is given by a momentum covector *P*_{a}. The associated mass is(6.6)The hyper-relativistic angular momentum of an elementary system is given by a tensor of the form(6.7)where the hyperspinor is required to be trace-free: . The angular momentum is defined with respect to a choice of origin in such a manner that under a change of origin defined by a shift vector *b*^{a} we have . In the case *r*=2 these formulae reduce to the usual expressions for relativistic momentum and angular momentum in a Minkowskian setting. The real covector(6.8)is invariant under a change of origin, and carries the interpretation of the intrinsic spin of the elementary system. The magnitude *S* of the spin is then defined by(6.9)In the case of a set of interacting hyper-relativistic systems we require that the total momentum and angular momentum should both be conserved. This then implies conservation of the total mass and spin.

## 7. Geometry of complex null-separation

In Minkowski space it is useful to examine the geometry of the space of complex null vectors at a point. Thus we take the case *r*=2 and consider complex vectors *z*^{a} satisfying *g*_{ab}*z*^{a}*z*^{b}=0. This implies that *z*^{a} can be written in the form . If we take a projective point of view, then the space of complex vectors at a point in Minkowski space can be regarded up to scale as a complex projective space . The null directions constitute a quadric in that space, which owing to the decomposition , has the structure of a doubly ruled surface . We can identify the first set of lines (the *α*-lines) with the projective unprimed spinors, and the second set of lines (the *β*-lines) with the projective primed spinors. The quadric has the property that two lines of the same type do not intersect, whereas two lines of the opposite type intersect at a point in —i.e. the null direction they together determine.

In the case of a general QST, we consider the space of complex vectors at a point of , and examine the corresponding space of directions, which has the structure of a complex projective space . The vanishing of the chronometric form identifies the space of complex null directions as a hypersurface of degree *r* in , which we shall call .

The points of correspond to ‘weakly’ null directions. The strongly null directions in , corresponding to those for which the associated null vectors are of minimal rank and hence of the form , constitute a subvariety defined by the mutual intersection of a system of quadrics, given by the equation *g*_{ab…c}*z*^{a}*z*^{b}=0. In this case, we have , and we can identify the two systems of (*r*−1)-planes by which is foliated, which we refer to as *α*-planes and *β*-planes, as the spaces of projective unprimed and primed hyperspinors, respectively.

The various null directions of intermediate degree correspond to points in lying on the linear spaces spanned by the joins of *k* points in (*k*=2, 3,…,*r*). The degree of nullness, as defined in §3, is given by the integer *k*.

In the case *r*=3, for example, the space of directions at a point in ^{9} is , and the null directions constitute a cubic hypersurface . The null directions of the first degree (i.e. the totally null directions) lie in the doubly foliated surface in ^{7}. The points of ^{7} all lie on the ‘first join’ of with itself; in other words, any point of ^{7} lies on a line joining two points of . The space then consists of null directions that are strictly of the second degree. (A null direction is strictly of the second degree if it is of the second degree but not also of the first degree.) Note that any point of can be represented as the join of three points in .

In the case *r*=4, the space of directions at a point in ^{16} is , and the null directions constitute a quartic hypersurface . The null directions of the first degree lie on the doubly foliated surface in ^{14}. The null directions of the second degree lie on the first join of with itself: . The null directions of the third degree lie in and constitute the general elements of ^{14}.

It is interesting to note a distinction between the case *r*=2 and higher-dimensional QSTs. In Minkowski space, the space of complex null directions at a point corresponds to a non-degenerate quadric in , which is doubly ruled in the sense that . In fact, any non-degenerate quadric in has this structure, and by an automorphism of one can transform one such quadric into another.

For *r*>2, this is generally not the case. For example, in the case *r*=3, the general cubic hypersurface in does not contain a doubly foliated subvariety . The space of cubic hypersurfaces in has a (complex) dimension 164. If we factor out by the group of projective automorphisms of , which is of dimension 80, then we are left with the 84 dimensional moduli-space for cubic forms in . The chronometric form of our QST geometry represents a single point in this moduli-space, and as such, constitutes a special cubic surface in . Such a hypersurface is, in fact, completely characterized by the existence of an embedded hyperspinorial subvariety of the form . Any two cubic hypersurfaces in admitting a hyperspinorial subvariety can be transformed into one another by an automorphism of , and thus represent the same point in the moduli-space. A similar observation applies for all *r*>2.

## 8. Generalized Klein representation

To proceed further it will be useful if we set the foregoing material in a geometric context that emphasizes the conformal properties of the chronometric form. To this end, we let denote the complex vector space of dimension 2*r* given by the pair . Let us write for a typical element of . Such an element will be referred to as a *hypertwistor*. For a brief introduction to the theory of hypertwistors (also called ‘generalized twistors’) see Hughston (1979). Let denote the space of dual hypertwistors. A natural pseudo-Hermitian structure can be introduced on the geometry of hypertwistors by means of the complex conjugation operation that maps to . The corresponding pseudo-Hermitian form is given by(8.1)and it is straightforward to verify that the inner product is invariant under the action of the group *U*(*r*,*r*).

The space of projective hypertwistors is a natural starting point for analysing the conformal geometry of complex QST, which can be regarded as the Grassmannian variety of projective (*r*−1)-planes in . More precisely, can be understood as the complex QST introduced earlier, together with some structure at infinity. Thus, is to be understood as a compactification of . The ‘finite’ points of correspond to the linear subspaces of that are determined by a relation of the form(8.2)for fixed *x*^{AA′}. The aggregate of such (*r*−1)-planes constitutes the points of . The (*r*−1)-planes for which *x*^{AA′} is Hermitian then constitute the points of the real space .

The conformal structure of QST is implicit in the various possibilities arising for the intersections of (*r*−1)-planes in hypertwistor space. A pair of (*r*−1)-planes in , in general, will not intersect. This general lack of intersection corresponds to the non-vanishing of the chronometric form for the corresponding QST points. In this connection, we note that the chronometric form introduced earlier for the pairs of real QST points is also well-defined for pairs of complex QST points. Now an (*r*−1)-plane in is represented by a simple skew hypertwistor *P*^{αβ…γ} of rank *r*. If *r*=2, we recover the fact that ordinary space-time points correspond to projective lines in , which in turn correspond to simple antisymmetric twistors of rank two. By a *simple* skew hypertwistor we mean one of the form for some collection *A*^{α}, *B*^{α},…,*C*^{α} of *r* hypertwistors (all of which lie on the given plane). Suppose that the simple skew hypertwistors *P*^{αβ…γ} and *Q*^{αβ…γ} represent, respectively, the (*r*−1)-planes *P* and *Q* in . Then, a necessary and sufficient condition for the vanishing of the chronometric form for the corresponding QST points is(8.3)where is the totally skew hypertwistor of rank 2*r*. We note that equation (8.3) is symmetric (*resp*., antisymmetric) under the interchange of *P*^{αβ…γ} and *Q*^{αβ…γ} if *r* is even (*resp*., odd). The vanishing of the form (8.3) is the condition that the projective planes *P* and *Q* contain a point in common. Equivalently, this means that the skew hypertwistors *P*^{αβ…γ} and *Q*^{ρσ…τ} contain at least one hypertwistor as a common factor. Thus we have deduced the following result.

*A necessary and sufficient condition for a pair of quantum space-time points to be weakly null-separated is that the corresponding* (*r*−1)-*planes in* *should intersect*.

More generally, the degree *k* of null separation for a pair of QST events is given by *k*=*r*−*m*−1, where *m* is the dimensionality of the intersection of the corresponding (*r*−1)-planes in . The possible degrees of null separation are given by *k*=1, 2,…,*r*−1. If we interpret (as usual) the case of no intersection as an intersection of dimension −1, then a non-null separation between the corresponding QST points can be interpreted as a ‘separation of degree *r*’. Thus, separations of degree less than *r* are all null, whereas a separation of degree *r* is non-null. The degree of separation of a pair of points, we recall, is given by the rank of the separation matrix . Equivalently, given two skew hypertwistors *P*^{αβ…γ} and *Q*^{αβ…γ}, each with *r* indices, let us form the dual hypertwistor by . Then *k* is given by the maximum number of index contractions we can make between *P*^{αβ…γ} and *Q*_{αβ…γ} without obtaining the result zero. If a single index contraction gives zero, this corresponds to the case where *P*^{αβ…γ} is proportional to *Q*^{αβ…γ}. Thus *k*=0 (separation of degree zero) can be interpreted as the ‘degenerate’ case where the two QST points coincide.

## 9. Quantum infinity

As indicated in the previous section, for any skew hypertwistor *P*^{αβ…γ} of rank *r* we define its dual *P*_{αβ…γ} by the relation . Here is the totally skew hypertwistor of rank 2*r*, which is unique up to an overall scale factor. Depending on whether *r* is even or odd, we have the following interchange relations: . Thus if *r* is even, then once the scale of the totally skew hypertwistor is fixed we obtain a *symmetric inner product* on the space of skew hypertwistors of rank *r*, which we can denote symbolically by(9.1)On the other hand, if *r* is odd then the inner product (9.1) is a symplectic structure. In this respect, the cases for even *r* and odd *r* are quite distinct. We shall return to this issue later when we specialize to the symmetric case.

Recall that under complex conjugation the skew hypertwistor *P*^{αβ…γ} becomes . If *P*^{αβ…γ} is simple, thus corresponding to an (*r*−1)-plane *P* in , then we say that *P* is a *real* plane if is proportional to *P*_{αβ…γ}. The real (*r*−1)-planes thus defined correspond to the real points of QST.

The points at infinity in the compactified space can be described as follows. In the projective hypertwistor space we choose a real (*r*−1)-plane *I* represented by a simple skew hypertwistor *I*^{αβ…γ}. The point *I* in corresponding to the (*r*−1)-plane *I* in will be called the *point at infinity*. The locus in consisting of all points that are chronometrically null-separated from *I* will be called *null infinity*. (There will be no ambiguity if we use the symbol *I* to denote both the point *I* in and the corresponding (*r*−1)-plane in .) It should be evident that null infinity has a rich structure, with various domains that can be classified according to their degree of null separation from the point *I*.

The ‘finite’ points of are those for which the chronometric separation from *I* is non-null, i.e. those points *P* for which 〈*P*,*I*〉≠0 with respect to the inner product (9.1).

*In the case of two finite quantum space-time points*, *the chronometric form Δ is given by the following ratio:*(9.2)

Equivalently, we can write . If *P* and *I* are not null separated, then we can choose the scales of *P*^{αβ…γ} and *I*^{αβ…γ} such that 〈*P*,*I*〉=1, without loss of generality, and similarly for *Q*^{αβ…γ} and *I*^{αβ…γ}. This leads to a further simplification in formula (9.2).

In general, even in the absence of such a simplification, we note that *Δ*(*P*,*Q*) is independent of the scale of *P*^{αβ…γ} and *Q*^{αβ…γ}. On the other hand, *Δ*(*P*,*Q*) does depend on the scale of and the scale of *I*^{αβ…γ}. It has an epsilon ‘weight’ of −1 and an *I* ‘weight’ of −2 (cf. Hughston & Hurd 1982). However, if we form the ratio associated with four hypertwistors *P*, *Q*, *R* and *S*, given by(9.3)where *p*, *q*, *r* and *s* are the QST points corresponding to *P*, *Q*, *R* and *S*, respectively, then we obtain an expression that is absolute—that is to say, a geometric invariant. This is because *Δ*(*P*,*Q*) has the ‘dimensionality’ of time raised to the power *r*; whereas the ratio (9.3) arises as a comparison of two such time intervals, and thus is dimensionless. The basic chronometric geometry, with infinity chosen as indicated above, admits no absolute or ‘preferred’ unit of time: in this geometry only ratios of time-intervals have an absolute meaning.

## 10. Cosmological infinity

There is, however, no reason *a priori* why such a ‘minimal’ structure should prevail at infinity. Other choices are available for *I*^{αβ…γ}, and these have the effect of giving the structure of a cosmological model. In the case *r*=2, for example, if *I*^{αβ} is chosen to be real and non-simple, then the quadratic form , which has an epsilon weight of one and an *I*-weight of two, has the dimensionality of inverse squared-time. Hence in this case there *is* a preferred unit of time.

To pursue this point further, we recall that ^{4} has the structure of a quadric *Ω* in . More specifically, for *r*=2 the space of skew rank-two twistors is , which is projectively , and ^{4} is the locus defined by the homogeneous quadratic equation . Infinity in ^{4} can then be defined by the intersection of ^{4} in with the projective 4-plane *I* given by the equation . If *I*^{αβ} is simple, then *I*^{4} is tangent to ^{4}, and the intersection is a cone—the null cone at infinity. On the other hand, if *I*^{αβ} is not simple, then the intersection *I*^{4}∩^{4} is a three-quadric. The resulting geometry, if *I*^{αβ} is real, is that of de Sitter space (Penrose 1967), and the parameter has the interpretation of being the associated *cosmological constant*. The de Sitter group then consists of those projective transformations of that preserve both *Ω* and the point *I*. In fact, with the incorporation of some additional structure at infinity, the entire class of Friedmann–Robertson–Walker cosmological models can be represented in this way (Penrose & Rindler 1984; Hurd 1985, 1995; Penrose 1995).

For general *r* a similar situation arises—in other words, the choice of structure at infinity gives rise in general to a chronometric geometry that is not flat, thus giving the character of a cosmological model. The key point is that, whereas in the case of a standard four-dimensional cosmological model based on Einstein's theory the existence of structure at infinity has a bearing on the geometry of space-time alone, in the case of a hypercosmology the structure at infinity also has implications for microscopic physics. In particular, while in the four-dimensional de Sitter cosmology the relevant structure at infinity contains the information of a single dimensional constant (the cosmological constant), in the higher-dimensional situation there will in general be a number of such dimensional constants emerging as geometrical invariants. Thus, within the same overall geometric framework one has the scope for introducing structure (or what amounts to the same thing—the breaking of symmetry) on both a global or cosmological scale, as well as on those scales of distance, time, and energy associated with the phenomenology of elementary particles. In order to prepare the groundwork necessary as a basis for investigating this idea in more detail we must now introduce the particular structure in needed to make its relation to ordinary four-dimensional space-time apparent.

## 11. Segré embedding and symmetry-breaking mechanism

Let us therefore consider the mechanisms for symmetry breaking at our disposal in the case of a standard ‘flat’ QST endowed with a canonical reality structure and null infinity. We shall demonstrate that the breaking of symmetry in a QST is intimately linked to the notion of quantum entanglement.

In practical terms the breaking of symmetry can be represented by an ‘index decomposition’. The point is that if the dimension *r* of the hyperspin space is not a prime number, then a natural method of breaking the symmetry arises by consideration of the decomposition of *r* into factors. The specific assumption that we shall make at this juncture will be that the dimension of the hyperspin space is *even*. Then we write *r*=2*n*, where *n*=1,2,…, and set , where * A* is a standard two-component spinor index, and

*i*will be called an

*internal*index (

*i*=1,2,…,

*n*). Thus we can write , where is a standard spin space of dimension two, and is a complex vector space of dimension

*n*. The breaking of the symmetry then amounts to the fact that we can identify the hyperspin space with the tensor product of these two spaces.

We shall assume, moreover, that there is a canonical anti-linear isomorphism between the complex conjugate of the internal space and the dual space . In other words, if , then we can write for the complex conjugate of *ψ*^{i}, where . Therefore, is a complex Hilbert space—and indeed, although for the moment we consider for technical simplicity the case for which *n* is finite, one should have in mind also the infinite-dimensional situation.

For the other hyperspin spaces we write , and , respectively. These equivalences preserve the duality between and , and between and ; and at the same time are consistent with the complex conjugation relations between and , and between and . Hence if then under complex conjugation we have , and if then .

In the case of a QST vector *r*^{AA′} we have a corresponding induced structure indicated by the identification(11.1)When the QST vector is real, the weak Hermitian structure on *r*^{AA′} is manifested in the form of a standard *weak* Hermitian structure on the spinor index pair, together with a *strong* Hermitian structure on the internal index pair. (In the case of a strong Hermitian structure it is assumed that there is a canonical isomorphism between the complex conjugate of the given complex vector space and the dual of that vector space.) In other words, the Hermitian condition on the QST vector *r*^{AA′} is given by .

One striking consequence of these relations is that we can interpret each point in QST as being a *space-time valued operator*. Ordinary classical space-time then ‘sits’ inside the QST in a canonical manner—namely, as the locus of those points of QST that factorize into the product of a space-time point *x*^{AA′} and the identity operator on the internal space(11.2)Thus, in those situations for which special relativity amounts to a satisfactory theory, we can regard the relevant events as taking place on or in the immediate neighbourhood of this embedding of the Minkowski space ^{4} in .

This picture can be presented in somewhat more geometric terms as follows. The hypertwistor space in the case *r*=2*n* admits a Segré embedding of the form . Many such embeddings are possible, although they are all equivalent to one another under the action of the overall symmetry group *U*(2*n*, 2*n*). If the symmetry is broken and one such embedding is selected out, then following the conventions discussed earlier we can introduce homogeneous coordinates and write *Z*^{α}=*Z*^{αi}. Here the bold greek letter * α* denotes an ordinary twistor index (

*=0, 1, 2, 3) and*

**α***i*denotes an internal index (

*i*=1, 2, …,

*n*). The Segré embedding consists of those points in for which we have a product decomposition of the associated hypertwistor given by .

The idea of symmetry breaking that we are putting forward here is related to the notion of disentanglement in standard quantum mechanics (cf. Gibbons 1992; Brody & Hughston 2001). That is to say, at the unified level the degrees of freedom associated with space-time symmetry are quantum mechanically entangled with the internal degrees of freedom associated with microscopic physics. The phenomena responsible for the breakdown of symmetry are thus analogous to the mechanisms of decoherence through which quantum entanglements are gradually diminished.

The compactified complexified space can be regarded as the aggregate of projective (2*n*−1)-planes in . Now, generically a in will not intersect the Segré variety . Such a generic (2*n*−1)-plane corresponds to a generic point in . The (2*n*−1)-planes that correspond to the points of can be constructed as follows. For each line *L* in we consider the subvariety where . For any algebraic variety (*j*≤*l*−1) we define the *span* of *V*^{j} to be the projective plane spanned by the points of *V*^{j}. More precisely, we say a point *X* in the ambient space lies in the span of the variety *V*^{j} if and only if there exist *m* points in *V*^{j} for some *m*≥2 with the property that *X* lies in the (*m*−1)-plane spanned by those *m* points. The dimension *k* of the span of *V*^{j} satisfies *j*≤*k*≤*l*; however, the value of *k* depends on the geometry of *V*^{j}.

Now the linear span of the points in , for any given *L*, is a (2*n*−1)-plane. This is the in that represents the point in corresponding to the line *L* in . The aggregate of such special (2*n*−1)-planes, defined by their intersection properties with the Segré variety ^{n+2}, constitutes a submanifold of , and this submanifold is compactified complexified Minkowski space. Thus we see that once the symmetry of has been broken in the particular way we have discussed, then Minkowski space can be identified as a submanifold.

## 12. Causality and quantum states

The embedding of Minkowski space in given by equation (11.2) implies a corresponding embedding of the Poincaré group in the hyper-Poincaré group. Indeed, if in ^{4} the standard Poincaré group consists of transformations of the form(12.1)then the hyper-Poincaré transformations in are of the form(12.2)On the other hand, with the identification *A*=* Ai*, the general hyper-Poincaré transformation in the broken symmetry phase can be expressed in the form(12.3)Thus the embedding of the Poincaré group as a subgroup of the hyper-Poincaré group is given by and .

Bearing these relations in mind, we now consider the problem of constructing a certain class of maps from the general even-dimensional QST to Minkowski space. It will be shown that under rather general and reasonable physical assumptions such maps necessarily take the form(12.4)where is a *density matrix*, that is to say, a positive semi-definite Hermitian matrix with unit trace. Thus the maps under consideration can be regarded as quantum expectations.

*Let* *satisfy the following conditions:* *(i) ρ is linear and maps the origin of* *to the origin of* ^{4}, *(ii) ρ is Poincaré invariant, and* *(iii) ρ is causal*. *Then ρ is given by a density matrix on the internal space*.

The general linear map from to ^{4} preserving the origin is of the form(12.5)where is weakly Hermitian. Now suppose that we subject to a Poincaré transformation of the form (12.2), and require the corresponding transformation of ^{4} to be of the form (12.1). If *ρ* satisfies these conditions then we shall say that the map *ρ* is Poincaré invariant. Thus Poincaré invariance holds if and only if(12.6)for all , all , and all . Thus we have(12.7)for all , and(12.8)for all *b*^{AA′}. Now equation (12.7) implies that *ρ* is of the form for some . Then, equation (12.8) implies that *ρ* must satisfy the trace condition . Finally, we require that if and are QST points such that the interval is future-pointing, then must also be future pointing, where(12.9)This is the requirement that *ρ* should be a causal map. However, this condition immediately implies that *ρ* must be positive semi-definite. The argument is as follows. If is future-pointing then it is necessarily of the form . Consider therefore the case for which is strongly null. Then we require that should be future-pointing (or vanish) for any choice of *α*^{Ai}. Thus in particular we require that should be future-pointing if *α*^{Ai} is of the form for any choice of *α*^{A} and *ψ*^{i}. This means that for all *ψ*^{i}, which shows that must be positive semi-definite. Since we have already shown that the trace of must be unity, it follows that is a density matrix. ▪

This theorem shows how the causal structure of QST is linked in a surprising way with the probabilistic structure of quantum mechanics. The concept of a quantum state emerges when we ask for consistent ways of ‘averaging’ over the geometry of QST in order to obtain a reduced description of phenomena in terms of the geometry of Minkowski space.

It is interesting to note that theorem 12.1 has a formal resemblance to Gleason's theorem in quantum mechanics, which states that a map from an observable to a real number (expectation value) must be a density matrix, if appropriate probabilistic conditions are imposed. In the present framework we see that a probabilistic interpretation of the map from a general QST to Minkowski space emerges as a consequence of elementary causality requirements. We can thus view the space-time events in as representing quantum observables, the expectations of which correspond to points of ^{4}.

## Acknowledgments

D.C.B. gratefully acknowledges financial support from The Royal Society. The authors are grateful to E. J. Brody for numerous ideas and suggestions in connection with the material presented here. We thank J. Butterfield, D. Finkelstein, G. W. Gibbons, R. Penrose, D. C. Robinson, and an anonymous referee, for helpful comments. Discussions with seminar participants at Oxford University, Imperial College London, and the Institute of Theoretical Physics, Wroclaw, have also been very helpful.

## Footnotes

- Received July 1, 2004.
- Accepted February 17, 2005.

- © 2005 The Royal Society