Spontaneously breaking space-time: Poincare,

gravity, cosmology, and spinning objects

w/ L. V. Delacretaz, A. Monin, R. Penco, and F. Riva (hep-th/1405.7384)

CERN Feb 27th 2015

Solomon EndlichEPFL

…or how I learned to stop worrying and love the coset construction

SSB of space-time symmetries + coset constructionGravity from the coset perspective

1)

2)

Example: Spinning object3)

Example: Cosmology4)

The Spontaneous breaking of space-time

symmetries

Act 1:

Observation:

Experiments

| i spontaneously breaks Poincaré

laws of physics Poincaré invariant!

Experience “stuff” is not

Consider the (point-like) Lobster:

x

µ = (t, 0, 0, 0)

Unbroken =

⇢P0 time translations

Jij spatial rotations

Broken =

⇢Pi spatial translations

J0i ⌘ Ki boosts

Consider the (point-like) Lobster:

⌘µ⌫dx

µ

d⌧

dx

⌫

d⌧

= �1⌘µ⌫dx

µ

d⌧

dx

⌫

d⌧

= �1

perturb w/ constraint:

�m

Zd⌧ = �m

Zd�

r�⌘µ⌫

dx

µ

d�

dx

⌫

d�

Zdt

✓�m+

1

2mv2 + ...

◆

usual treatment:

Consider the (point-like) Lobster:

Is there a way to construct the theory for the Goldstones (fluctuations) based on the symmetry breaking pattern alone?

Almost!*

*Ivanov and Ogievetsky 1970’s

more recent literature:Neilson and Chadha (1976)

Schäfer, Son, Stephanov, Toublan, and Verbaarschot hep-ph/0108210 (2001)

Low and Manohar hep-th/0110285 (2002)

Watanabe and Brauner 1109.6327 (2011)

Nicolis, and Piazza 1112.5174 (2011)

Watanabe and Murayama 1203.0609 (2012)

Hidaka 1203.1494 (2012)

Nicolis, Penco, Piazza, and Rosen 1306.1240 (2013)

Endlich, Nicolis, and Penco 1310.2272 (2013)

Endlich, Nicolis, and Penco 1311.6491 (2013)

Brauner, Endlich, Monin, and Penco 1407.7730 (2014)

Nicolis, Penco, Piazza, and Rattazzi 1501.03845 (2015)

etc.

Why “almost” (classic interpretation)?

�O(x) = i

X

I

⇡I(x)TIhO(x)i

not independent fluctuations

X

I

⇡I(x)TIhO(x)i = 0?

eliminate redundancy via“inverse Higgs constraint”

NBroken Generators

NGoldstones6=

one usually hears that:

Ex.Unbroken =

⇢P0 time translations

Jij spatial rotations

Broken =

⇢Pi spatial translations

J0i ⌘ Ki boosts

time-dependent boost time-dependent translation

Vs

same information⇡

boost

⇠ d

dt⇡trans

⇡boost

K + ⇡trans

P = 0

(however…can have no redundancy: choice?)

MGoldstone 6= 0 (read: )gap

⇤strong

µ

Energy

A. Nicolis, R. Penco, F. Piazza, and R. Rosen (hep-th/1306.1240)

…or here

work here…

Mechanism dependent (sort of)S. Endlich, A. Nicolis, and R. Penco (hep-th/1311.6491)

basic construction: 1

1) identify symmetry breaking pattern G ! H

2) build objects out of Goldstones that transform linearly*

*along the lines of Callan, Coleman, Wess and Zumino + Volkov

D⇡ ! h(g,⇡)D⇡

basic construction: 2

4) build a Lagrangian out of H invariant objects made up of D⇡a

3) Inverse Higgs constraints (algebra) [P ,X] = X 0 + ...

=> structure of covariant derivatives

can impose: should? necessary? depends…

D⇡ = 0 =) ⇡(@⇡0)

Expanding the magical step 2)

CCWZ construction generalized to space-time symmetries:

⌦(y,⇡) ⌘ eiya(x)Paei⇡

↵(x)X↵parametrize the coset

g⌦(y,⇡) = ⌦(y0,⇡0)h(y,⇡, g)transformation rules

X↵ = broken generators

¯Pa = unbroken translations

TA = other unbroken generators

GorganizeH0

Expanding the magical step 2) … part 2

Maurer-Cartan form ⌦�1d⌦⌦�1@µ⌦ = Eµ

a(Pa +ra⇡↵X↵ +AB

a TB)

ra⇡↵(x)

g�! ra⇡0↵(x) = ha

b(y,⇡, g)h�↵(y,⇡, g)rb⇡

�(x)

transforms as a covariant derivative

transforms as a connection

rHa ⌘ [(E�1)a

µ@µ + iABa TB ]

transforms as a vierbeind

4x detE

Gravity from the coset perspective

Act 2:

rule of thumb (CCWZ):

Gauge symmetries

⌦�1@µ⌦ ! ⌦�1Dµ⌦ ⌘ ⌦�1(@µ + iAIµVI)⌦.

can we do the same for space-time symmetries? yes

Gravity as “gauged ISO(3,1) with non-linearly

realized translations”

⌦ ⌘ eiya(x)Pa

1) non-linearly realized translations

2)

⌦�1Dµ

⌦ ⌘ e�iy

a(x)Pa

✓@µ

+ ieµ

aPa

+i

2!ab

µ

Jab

◆eiy

a(x)Pa

= ieµ

aPa

+i

2!ab

µ

Jab

eµa = @µy

a + eµa + !ab

µ yb vierbein

“spin” connection

all we have is:

rLa ⌘ (e�1)a

µ(@µ +i

2!bcµ Jbc)

4) S =

Zd

4x det eL(rL

a ) GR?

[rLa ,rL

b ]Vc = Rc

dabVd � Tab

drLd V

c

S = � 1

16⇡G

Zdet(e)d4x

"R

abab +

3X

i

ciT2 + · · ·

#

�S

�!abµ

= 0

Tabc = 0 torsion free!

!(e) ⇠ e�1@e+ ...

GR! (as EFT)

Spinning object

Act 3:

1) SSB pattern:

S ✓ SO(d)with

Unbroken =

⇢P0 time translations

¯Jij = Sij + Jij residual rotations

Broken =

⇢Pi spacial translations

Jab rotations and boosts

Utilizing the coset

2) with ⌦ = eiyaPaei↵abJ

ab/2

x

µ⌦�1Dµ⌦ = iE(P0 +r⇡

iPi +

1

2r↵cdJ

cd)

E = x

⌫e⌫

a⇤a0

r⇡

i = E

�1x

⌫e⌫

a⇤ai

r↵

ab = E

�1⇣⇤ ac ⇤cb + x

µ!µ

cd⇤ca⇤d

b⌘

=d⌧

d�

⇤(↵) = ⇤(⌘)⇤(⇠)with

3) as⇥P0,Ki

⇤⇠ Pi r⇡i = 0

ua⇤ai(⌘) = 0

boost into rest frame accompanying rotating

object

Utilizing the coset

4) build the action

S =

Zd�E

✓�m+

Iijkl4

r↵ijr↵kl + · · ·◆

EFT describing rotating object coupled to gravity

(as advertised)

r↵ = ⇤�1d⌧⇤+ uµ⇤�1!µ⇤with

Utilizing the coset

Turn OFF gravity… what do we get?

Reality check

�

�⇠i(· · · ) = 0 Euler equations

Zdt

1

2Iij⌦

i⌦j + · · · correct leading kinetic energy

What about the ’s? · · ·

Real objects are NOT perfectly rigid… they distort under stress

Meaning of higher derivative terms

Dimensionally Sr↵4 ⇠ S ⌦4 ⇠✓⌦2

!20

◆(I⌦2)

For some “elastic” object EFT tells you related to modes we have integrated out… the

!0

normal modes!

“spherical chunk” I ⇠ �“cubic crystal” S ⇠ �� + cubic

@t⌦ =I�1 ((I⌦)⇥ ⌦) + I�1 ((S⌦⌦⌦)⇥ ⌦)

+ I�1(S⌦⌦) ((I⌦)⇥ ⌦) +O �S⌦2/I

�2

here

Precession!

@t⌦ = I�1 ((S⌦⌦⌦)⇥ ⌦) +O �S⌦2/I

�2

Generalized Euler (in 3 d):

Spherical chunk of cubic crystal rotating off axisEx:

precession of angular velocity (as viewed in body frame)

-1

0

1

-1

0

1

1.8

1.9

2.0

… for some random input values

is this known?

Additional vertices, such as

Coupling to gravity

1

4MPlLk✏kij@ih0

j

⇠ 1

M2Pl

Lk✏kij⇥hi�@�h0

j + · · ·⇤

where L = ⇤(⇠)�I⌦+ S⌦3 + · · ·

�is the angular momentum

Compute (spin-orbit, spin-spin, etc.)

(À la Goldberger, Porto, Rothstein, and others)

cosmological theories

Act 4:

EFT of InflationThe early universe: homogenous, isotropic, and slightly time dependent

Inflation driven by

Time-translations spontaneously broken

Construct a systematic effective field theory

(Creminelli, Luty, Nicolis, Senatore 2006Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore 2007)

Goldstone boson = adiabatic perturbations

�a = �a(t)

⇥a = ⇥a(t+ �(x)) � ⇥a(t) + ⇤t⇥a(t) · �(x)

Can we write down action for Goldstones directly?

Proceed by gauging the space-time symmetries of de Sitter

Couple to matter that spontaneously breaks some of these symmetries (+ internal)

Construct a systematic effective field theory

spontaneously broken TIME + internal shifts

spontaneously broken SPACIAL SHIFTS + internal shifts

EFT of inflation

Solid Inflation

Ex.

Conclusions“stuff” non-linearly realizes space-time symmetries (usual procedures don’t work without being clever)

in EFT one needs to identify 1) d.o.f. and 2) symmetries: coset helps with both

general procedure to couple the Goldstones to gravity

straightforward and exhaustive procedure

SPIN: made simpler? new expansion parameter to assist in computations

straightforward construction of inflation

Applications: cosmology, plasma physics, exotic condensed matter states, etc. all in a model independent fashion

(!/!0)