Rotation Group, the Holographic Wave Function

This chapter discusses everything in terms of wave functions and describes the choice between active and passive rotations. It defines the representations of the rotation group by means of the abstract state vectors and by means of wave functions. That implies that the rotations are active rotations. The chapter finds that for a sequence of rotations the three-dimensional matrices have the property. It considers the change of the wave function in active and passive rotations respectively. In the active interpretation the physical system — characterized in space by the wave function — is bodily rotated. In the passive interpretation the frame of reference is rotated, whereas P and the local physical state at P remain unchanged. The chapter describes that there is a description which uses the new coordinates but describes the old situation.This chapter discusses everything in terms of wave functions and describes the choice between active and passive rotations. It defines the representations of the rotation group by means of the abstract state vectors and by means of wave functions. That implies that the rotations are active rotations. The chapter finds that for a sequence of rotations the three-dimensional matrices have the property. It considers the change of the wave function in active and passive rotations respectively. In the active interpretation the physical system — characterized in space by the wave function — is bodily rotated. In the passive interpretation the frame of reference is rotated, whereas P and the local physical state at P remain unchanged. The chapter describes that there is a description which uses the new coordinates but describes the old situation.

CRC Press eBooksJul 14, 20230 citations

Unitary networks from the exact renormalization of wave functionals

Jackson R. Fliss, Robert G. Leigh, Onkar Parrikar

The exact renormalization group (ERG) for $O(N)$ vector models (at large $N$ ) on flat Euclidean space can be interpreted as the bulk dynamics corresponding to a holographically dual higher spin gauge theory on ${\mathrm{AdS}}_{d+1}$ . This was established in the sense that at large $N$ the generating functional of correlation functions of single-trace operators is reproduced by the on-shell action of the bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. In this paper, we extend the ERG formalism to the wave functionals of arbitrary states of the $O(N)$ vector model at the free fixed point. We find that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Consequently, the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and its continuum version, cMERA, which have appeared recently in holographic contexts. In particular, the ERG tensor network appears to share the general structure of cMERA but differs in important ways. We comment on possible holographic implications.The exact renormalization group (ERG) for $O(N)$ vector models (at large $N$) on flat Euclidean space can be interpreted as the bulk dynamics corresponding to a holographically dual higher spin gauge theory on ${\mathrm{AdS}}_{d+1}$. This was established in the sense that at large $N$ the generating functional of correlation functions of single-trace operators is reproduced by the on-shell action of the bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. In this paper, we extend the ERG formalism to the wave functionals of arbitrary states of the $O(N)$ vector model at the free fixed point. We find that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Consequently, the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and its continuum version, cMERA, which have appeared recently in holographic contexts. In particular, the ERG tensor network appears to share the general structure of cMERA but differs in important ways. We comment on possible holographic implications.

arXiv:1609.03493Physical review. D/Physical review. D.Jun 6, 201721 citations

The exact renormalization group (ERG) for

O(N)

vector models (at large

N

) on flat Euclidean space can be interpreted as the bulk dynamics corresponding to a holographically dual higher spin gauge theory on

{\mathrm{AdS}}_{d+1}

. This was established in the sense that at large

N

the generating functional of correlation functions of single-trace operators is reproduced by the on-shell action of the bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. In this paper, we extend the ERG formalism to the wave functionals of arbitrary states of the

O(N)

vector model at the free fixed point. We find that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Consequently, the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and its continuum version, cMERA, which have appeared recently in holographic contexts. In particular, the ERG tensor network appears to share the general structure of cMERA but differs in important ways. We comment on possible holographic implications.