On the quantisation of gravity by embedding spacetime in a higher dimensional space

Abstract

Abstract. Certain difficulties of quantum gravity can be avoided if we embed the spacetime V 4 into a higher dimensional space V N ; then our spacetime is merely a 4-surface in V N . What remains is conceptually not so difficult: just to quantise this 4-surface. Our formal procedure generalises our version of Stueckelberg's proper time method of worldline quantisation. We write the equations of V 4 in the covariant canonical form starting from a model Lagrangian which contains the classical Einstein gravity as a particular case. Then we perform quantisation in the Schrödinger picture by using the concepts of a phase functional and wave functional. As a result we obtain the uncertainty relations which imply that an observer is 'aware' either of a particular spacetime surface and has no information about other spacetime surfaces (which represent alternative histories); or conversely, he loses information about a particular V 4 whilst he obtains some information about other spacetimes (and histories). Equivalently, one cannot measure to an arbitrary precision both the metric on V 4 and matter distribution on various alternative spacetime surfaces. We show how this special case in the 'coordinate' representations can be generalised to an arbitrary vector in an abstract Hilbert space.

Rationale for a correlated worldline theory of quantum gravity

“…One thing that has been learnt from such efforts is quite how hard it is to change GR without finding either instabilities, or acausal behaviour, or a breakdown of unitarity caused by ghost modes (a nice example being 'massive gravity' theories [80], where a constraint removes the Boulware-Deser ghost but then makes the theory acausal [81]). Related lines of thought look at 'induced gravity' [82], or more general 'emergent gravity' [83] scenarios; and there is a long tradition, recently revived, of looking for extended versions of classical GR embedded in higher dimensional spaces [84,85].…”

p-BRANES, POISSON-SIGMA MODELS AND EMBEDDING APPROACH TO (p+1)-DIMENSIONAL GRAVITY

“…This idea generated renewed interest in the search for a dynamical basis regarding the old embedding method [3,4]. Since then this approach [5] has been developed in several papers [6,7,8,9], where e.g. in [6] the model for gravity provided by a free bosonic p-brane action in curved (but conformally flat) target space-time was considered.…”

“…It should be stressed that at the classical level p-brane (and D p -brane) theories in these approaches have only matter (and gauge field) degrees of freedom, with gravity involving only auxiliary non-propagating ones. But a kinetic term for gravity should appear in the effective action of the quantum theory after the integration over the matter fields, yielding propagating gravity as a result of quantum effects (see, for example, the discussion of this point in [6]). Hence all the embedding models could be regarded as particular realizations of the concept of gravity induced by quantum corrections [10,11].…”

“…Hence the model can be regarded as a realization of the idea by Regge and Teitelboim for a 'string-like' description of gravity. In fact, some realizations of this idea were proposed previously [6,7,8,9]. So in [6] a model for gravity was constructed from the free bosonic p-brane action in curved (but conformally flat) space-time.…”

“…In fact, some realizations of this idea were proposed previously [6,7,8,9]. So in [6] a model for gravity was constructed from the free bosonic p-brane action in curved (but conformally flat) space-time. 4 The basic difference of our result with respect to [6] consists in the possibility to consider flat target space (i.e.…”