# Existence and construction of Galilean invariant $z\ne 2$ theories

## Abstract

We prove a no-go theorem for the construction of a Galilean boost invariant and $$z{\ne}2$$ anisotropic scale invariant field theory with a finite dimensional basis of fields. Two point correlators in such theories, we show, grow unboundedly with spatial separation. Correlators of theories with an infinite dimensional basis of fields, for example, labeled by a continuous parameter, do not necessarily exhibit this bad behavior. Hence, such theories behave effectively as if in one extra dimension. Embedding the symmetry algebra into the conformal algebra of one higher dimension also reveals the existence of an internal continuous parameter. Consideration of isometries shows that the nonrelativistic holographic picture assumes a canonical form, where the bulk gravitational theory lives in a space-time with one extra dimension. This can be contrasted with the original proposal by Balasubramanian and McGreevy, and by Son, where the metric of a ($d+2$)-dimensional space-time is proposed to be dual of a $d$-dimensional field theory. We provide explicit examples of theories living at fixed point with anisotropic scaling exponent $$z=\frac{2{\ell}}{{\ell}+1}$$, $${\ell}{\in}\mathbb{Z}$$.

- Authors:

- Univ. of California, San Diego, CA (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Univ. of California, San Diego, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1441112

- Alternate Identifier(s):
- OSTI ID: 1498885

- Grant/Contract Number:
- SC0009919

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 12; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; quantum field theory; continuous symmetries; spacetime symmetries; symmetries; group theory

### Citation Formats

```
Grinstein, Benjamín, and Pal, Sridip. Existence and construction of Galilean invariant z≠2 theories. United States: N. p., 2018.
Web. doi:10.1103/physrevd.97.125006.
```

```
Grinstein, Benjamín, & Pal, Sridip. Existence and construction of Galilean invariant z≠2 theories. United States. doi:10.1103/physrevd.97.125006.
```

```
Grinstein, Benjamín, and Pal, Sridip. Mon .
"Existence and construction of Galilean invariant z≠2 theories". United States. doi:10.1103/physrevd.97.125006.
```

```
@article{osti_1441112,
```

title = {Existence and construction of Galilean invariant z≠2 theories},

author = {Grinstein, Benjamín and Pal, Sridip},

abstractNote = {We prove a no-go theorem for the construction of a Galilean boost invariant and $z{\ne}2$ anisotropic scale invariant field theory with a finite dimensional basis of fields. Two point correlators in such theories, we show, grow unboundedly with spatial separation. Correlators of theories with an infinite dimensional basis of fields, for example, labeled by a continuous parameter, do not necessarily exhibit this bad behavior. Hence, such theories behave effectively as if in one extra dimension. Embedding the symmetry algebra into the conformal algebra of one higher dimension also reveals the existence of an internal continuous parameter. Consideration of isometries shows that the nonrelativistic holographic picture assumes a canonical form, where the bulk gravitational theory lives in a space-time with one extra dimension. This can be contrasted with the original proposal by Balasubramanian and McGreevy, and by Son, where the metric of a ($d+2$)-dimensional space-time is proposed to be dual of a $d$-dimensional field theory. We provide explicit examples of theories living at fixed point with anisotropic scaling exponent $z=\frac{2{\ell}}{{\ell}+1}$, ${\ell}{\in}\mathbb{Z}$.},

doi = {10.1103/physrevd.97.125006},

journal = {Physical Review D},

number = 12,

volume = 97,

place = {United States},

year = {2018},

month = {6}

}

DOI: 10.1103/physrevd.97.125006

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Works referencing / citing this record:

##
The lagrangian formulation of chiral scalars

journal, July 1987

- Imbimbo, C.; Schwimmer, A.
- Physics Letters B, Vol. 193, Issue 4

##
Warped conformal field theory as lower spin gravity

journal, August 2015

- Hofman, Diego M.; Rollier, Blaise
- Nuclear Physics B, Vol. 897

##
A new formulation of non-relativistic diffeomorphism invariance

journal, October 2014

- Banerjee, Rabin; Mitra, Arpita; Mukherjee, Pradip
- Physics Letters B, Vol. 737

##
Schrödinger holography for *z* < 2

journal, January 2015

- Andrade, Tomás; Keeler, Cynthia; Peach, Alex
- Classical and Quantum Gravity, Vol. 32, Issue 3

##
Localization of the Galilean symmetry and dynamical realization of Newton–Cartan geometry

journal, January 2015

- Banerjee, Rabin; Mitra, Arpita; Mukherjee, Pradip
- Classical and Quantum Gravity, Vol. 32, Issue 4

##
Schrödinger holography with *z* = 2

journal, March 2015

- Andrade, Tomás; Keeler, Cynthia; Peach, Alex
- Classical and Quantum Gravity, Vol. 32, Issue 8

##
Comments on string theory backgrounds with non-relativistic conformal symmetry

journal, October 2008

- Maldacena, Juan; Martelli, Dario; Tachikawa, Yuji
- Journal of High Energy Physics, Vol. 2008, Issue 10

##
Holographic stress tensor for non-relativistic theories

journal, September 2009

- Ross, Simon F.; Saremi, Omid
- Journal of High Energy Physics, Vol. 2009, Issue 09

##
Self-gravitating darkon fluid with anisotropic scaling

journal, November 2010

- Stichel, P. C.; Zakrzewski, W. J.
- The European Physical Journal C, Vol. 70, Issue 3