# Spacetime Symmetry Breaking

Daniel Naegels, An introduction to Goldstone boson physics and to the coset construction, arXiv: 2110.14504.

## Geometric Coset Structure

Many of the spacetime symmetries are non compact – e.g. dilatation, translations, boosts, etc. This indicates the useful group theory properties are not anymore systematically verified. Fortunately, the coset construction can be related to differential geometry, where non compact group could be developed.

Coset structure and General relativity analog. Yi-Ning

## spontaneous breaking of the conformal group to the Poincare group

D. V. Volkov, “Phenomenological Lagrangians,” Fiz. Elem. Chast. Atom. Yadra 4 (1973) 3–41.

E. A. Ivanov and V. I. Ogievetsky, “The Inverse Higgs Phenomenon in Nonlinear Realizations,” Teor. Mat. Fiz. 25 (1975) 164–177.

studied the spontaneous breaking of the conformal group down to the Poincare group.

R. Penco, “An Introduction to Effective Field Theories,” arXiv:2006.16285 [hep-th].

V. I. Ogievetsky, Nonlinear realizations of internal and space-time symmetries. PhD thesis, In X-th winter school of theoretical physics in Karpacz, Poland, 1974.

A. Salam and J. A. Strathdee, “Nonlinear realizations. 2. Conformal symmetry,” Phys. Rev. 184 (1969) 1760–1768.

## Condensed matter application

With an explicit computation of the oscillation modes of the lattice, it can be noticed that the number of NG modes are linked to the breaking of translations and that the rotations do not provide additional massless excitations.

H. Watanabe and H. Murayama, “Redundancies in Nambu-Goldstone Bosons,” Phys. Rev. Lett. 110 no. 18, (2013) 181601, arXiv:1302.4800 [cond-mat.other].

I. Low and A. V. Manohar, “Spontaneously broken space-time symmetries and Goldstone’s theorem,” Phys. Rev. Lett. 88 (2002) 101602, arXiv:hep-th/0110285.

T. Brauner and H. Watanabe, “Spontaneous breaking of spacetime symmetries and the inverse Higgs effect,” Phys. Rev. D 89 no. 8, (2014) 085004, arXiv:1401.5596 [hep-ph].

## Dilatation Symmetry Breaking

it is the NG mode associated to the breaking of dilatation while the breaking of the special conformal transformations is not providing additional NG modes.

the single dilaton associated to the breaking of dilatation symmetry and of special conformal transformation symmetries

Cosmological dilatation.