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Anisotropic Mechanics
J.M. Romero, V. Cuesta, J.A. Garcia, and
J. D. Vergara
Instituto de Ciencias Nucleares, UNAM, Mexico.
Introduction
Action Principle
Canonical Formalism
Equations of Motion
Dynamical Symmetries
Massless case, nonlinear realization of the
generators
Thermodynamic Properties
Conclusions
Introduction
In recent years, a new class of anisotropic scaling transformations has
been considered in the context of critical phenomena {Henkel}, string
theory {Son, Barbon, Peet, …} and quantum gravity{Horava,…}
t  bzt,
where
x  bx.
(1)
z plays the role of a dynamical critical exponent.
In particular the result of Horava is quite interesting since it produces a
theory that perhaps reduces to general relativity at large scales and
may provide a candidate for a UV completion of general relativity.
In the IR these systems approach the usual General
Relativity with local Lorentz invariance but in the UV the
formulation admit generalized dispersion relations of the
form
P02  G( P2 ) z  0,
G  const.
As the dispersion relations used by these models are quadratic in
while the spatial momentum scale as z the models are in principle
renormalizable by power counting arguments at least for z  3 .
(2)
P0
In the Horava theory, one hopes to obtain a higher spatial
derivative theory of gravity.
The idea of the renormalizability of this theory is the following
{Germani, et al.}. Consider the linearized gravity
g      h .
In the case of the Einstein-Hilbert action, this can be
expanded as
S
1
2
2
d
4
x[(h)  (h) h  ...]
2
2
Where the second term is a typical self-interaction term and is the
gravitational coupling constant, with mass dimension [ ]  1.
For perturbative renormalization, one defines
f  h/
Expanding around the fixed point we get
1
S   d 4 x[(f ) 2   (f ) 2 f  ...].
2
Since  has dimensions of length one expects that
.
  1/ E ,
Quantum mechanically the gravitational interaction is
irrelevant and this implies that the self-coupling spoils
the renormalizability at the Gaussian fixed point.
In the Horava theory, one hopes to obtain a higher spatial
derivative theory of gravitation, but only quadratic in time
1
S  2  d 4 x  h 2   i h i h   2( z 1) ( i1 ... iz h) 2  (h 2   i h i h)h  ...,
2
The coupling constant for the higher spatial derivative
part has negative mass units [ ]  1.
Now, using the redefinitions
h  
1 z
2
f
and
t   1 z t .
We get,
1 4
S   d x   f 2  1 i f  i f  ( i1  iz f ) 2  2 ( f 2  1 i f  i f ) f  
2
with
1   2(1 z )
and
2  
1 z
2
These parameters have now positive mass dimension.
Then, the dangerous irrelevant operator of GR has now been turned
into a relevant operator at the Lifshitz point.
Action Principle
Let us consider a d+1-dimensional space-time, for a
parametrized non-relativistic particle,
 m x2

S   d 
 V ( x)t ,
 2 t

These system has the following constraint:
2
p
  pt 
 V ( x).
2m
To obtain a dispersion relation of the Horava type, we can
start from the action
z


2 2( z 1)
 m x 

S   d L   d 
V  x t  ,
1
 2  t  z 1



(3)
where:
dxi
dt
t
, xi 
, x 2  xi x i , i  1, 2,
d
d
, d.
This action is also invariant under reparametrization, i.e.:
   ( f )
With the coordinates and the time are scalars under the
reparametrization.
Furthermore, if we consider that the potential:
V (bx )  b  zV ( x ), for example
a
V (x) 
,
z
|x|
a  constant
The action (3) will be invariant under the anisotropic scale
transformations (1).
Now, for z  2 and in the gauge
t 
we obtain
m 2
a 
S   dt  x  2  ,
|x| 
2
For arbitrary z results:
 m 2 2( zz1)
a
S   dt   x 
 z
|x|
2

,

Introducing an einbein the action (3) has the form
z


2
2
m
(x )

.
S   d

(
z

2)
e
z 2

2( z  1) 
e
t




Now, in the limit,
z  1, and at the same time m  0,
m
 1,
z 1
2
1

2 1/ 2  e 
S   d   x     e 
2
t 

1
Canonical Formalism
Canonical momenta
2 z
2 2( z 1)
L  mz  ( x )
pi  i  

1
x  z  1 
 t  z 1
xi ,
z


2 2( z 1)
L
 m (x )

pt   

V
(
x
)
 H ,
z

t
 2( z  1)  t  z 1



This theory has a constraint
 mz 
m
  pt 


2( z  1)  2( z  1) 
z
z
2
2
p
   V ( x )  0. (5)
Which can be rewritten, for V(x)=0, in the form
 pt 
2
G p
2

z
1
G  2z
z
 0,
 m 


2(
z

1)


2(1 z )
.
In this way, we get a dispersion relation that is very similar to
Horava’s gravity model. Notice that in the limit z  1 and m  0,
1
2 2
1  pt  ( p )  V ( x ).
Following Dirac, we propose at the quantum level that the
physical sates of the theory are invariant under the same
transformations,
iˆ
e

P

P
iˆ
e 
P

P
In infinitesimal form we get
ˆ 
P
0
In the coordinates representation we obtain

pt  i
t

pi  i i
q
z

 
2 2 2
i
  G      V ( x )  .
t 

So, we get a generalization of the Schroedinger Equation.
Gauge of freedom
Because of the action (3) is invariant under
reparametrizations, there is a local symmetry generated
by the constraint  . So the hamiltonian action is
invariant under the following gauge transformations:
z 2
2 2
V
 piò  { pi , ò}  ò ,
xi
 xiò  {xi , ò}  ò G z ( p )
pi ,
 tò  ò{t , }  ò,  ptò  0,
ò
ò  .

Equations of motion
gij x    x
j
with the metric is
z 2
2 2( z 1)

2( z  1) V
,
mz xi

2  z xi x j
gij    ij 
2
z

1
x

and the inverse

.

i j

xx
ij
ij
g     (2  z ) 2
x


,

Then, we can rewrite these equations in the form
xi x j
2(1  z ) 2 2(zz21) 
xi 
x 

  ij  ( z  2) 2
mz
x

 V
.

 x j
Dynamical Symmetries
Dynamical symmetries of the Schroedinger equation
H temporal translations
i
ij
kl
ik jl
jl ik
il jk
jk il
J
,
J

(

J


J


J


J ),


ij
J rotations
ij
k
ik
j
jk i
J
,
P

(

P


P ),


D dilatation
ij
k
ik
j
jk
i
J
,
K

(

K


K
),


M mass operator
i
i
H
,
K

P
  ,
P spatial translations
i
K Galilean boost
i
j
ij
P
,
K



M.


C special conformal transformations
For the Dilatations we have the algebra:
i
i
D
,
P

P
  ,
i
i
D
,
K

K
  ,
D, H   2H ,
and the Special Conformal Transformations:
D, C  2C , H , C  D,
C, P   K .
i
i
It is interesting to notice that H , C , D close themselves in a
SL (2, R ) subalgebra.
So, the Schroedinger algebra Schr (d ) is the Galilean
algebra plus dilatation and spatial conformal transformations.
We are interested in the construction of the explicit generators
of the relative Poisson-Lie Schroedinger algebra for any z.
We will denote the generalization of the Schroedinger
symmetry algebra for any z as Schrz (d ) in d dimensions. The algebra
is given by the Galilean algebra (29) plus
i
i
D
,
P

P
  ,
i
i
D
,
K

(1

z
)
K
,
 
D, M   (2  z )M , D, H    zH
Notice that M does not play the role of the center anymore
unless z=2 and that C, the generator of the Special
Conformal transformations is not in the algebra. For z=2
the phase space realization reads
J  x p  x p , H   pt , P  p ,
ij
i
j
j
i
i
i
K i   Mxi  tp i ,
1 2
C  t pt  tx p  Mx , D  2tpt  x i p i
2
2
i
i
Inspired by this construction we will display a set of
dynamical symmetries of the Schroedinger equation for
any z. The crux of the argument is to allow M to depend
2
on the magnitude of p squared M ( p )
M i
2
2 
p

2

M
,
M
(
p
)

A
(
p
) ,
i
p
where A is a constant. Then taking as our Schroedinger
equation for any z we have
2
p
S  pt  H  pt 
2
2M ( p )
(10)
we will ask for the phase space quantities that commutes
(in Poisson sense) with S
O, S  0,
(11)

is fixed in terms of the dynamical exponent z by
relating our definition (10) with the first class constraint
(5)

2 z
,
2
A
1
2 G
To find what are the analogs of D, C and K we propose
D  tpt   x i pi ,
C  t 2 pt   tx i pi   M ( p 2 ) x 2 ,
K i  tp i   M ( p 2 ) x i
and using (11) we obtain
1

,
2(1   )
1
 
,
(1   )
2
 
(1   ) 2
So we have the non trivial dynamical symmetries of the
anisotropic Schroedinger equation generated by
D  ztpt  x i pi ,
(12)
z2 2
z i
1
2
2
C  t pt  tx pi  M ( p ) x ,
4
2
2
and d Galilean boosts
z i
i
K  tp  M ( p 2 ) x i .
2
(13)
Unfortunately the dynamical symmetries given by (12), (13) and
plus J, P, H, M do not close. Nevertheless a subalgebra formed by H,C,D
indeed close into a SL(2,R) sector of the full algebra,
D, H    zH ,
D, C   zC ,
z
 H , C  D,
2
Massless case. Nonlinear realization
Consider the action
e 2
 0
i
2 2 
S   d  x p0  x pi   p0  f p  
2


where
f  f ( p 2 ).
Then, if it is possible to find a canonical transformation from
this action to the relativistic particle action,
e 2
 0
i
2 
S   d  X P0  X Pi   P0  P  
2


The symmetries of the Horava particle, will be exactly the
same that of the massless relativistic particle.
Such, canonical transformation exists and is given by
P0  p0 ,
X 0  x0 ,
Pi  fpi ,
i
i

j
2 f ' p pj  j
i
X   2
x .

2
 f

f

2
ff
'
p


where
f ( p )
f '
,
2
p
2
Then, for the new action we have the massless Klein-Gordon
equation and in consequence full conformal symmetry
SO(d+1,1),
with generators given by
J ab
and defined by
J   X  P  X P ,
J 1,0  D  X  P ,
1
J 1,    P  K   ,
2
1
J 0,    P  K   ,
2
where K  is the generator of the Spatial conformal
transformations.
K  2X  X  P  X 2 P
So, the Horava massless particle have the same
symmetries than the masless relativistic particle and
corresponds only to a nonlinear relization of the generators.
Thermodynamic Properties
In the following we shall sketch some of the thermodynamic
properties of the system. We will consider only the case
without potential. In this case the energy of our model is
H  G pz.
The canonical partition function for the system in a space of
d dimensions is
Z
V
N
 d pe
 G pz

Vd
N


0
d pp
d 1   G p z
e
.
z
Using u   G p , and the definition of the Gamma
function we obtain
Vd
Z
z
N
 G 
d
z
d 
  .
z
In this way, if N is the number of particles, the Helmholtz
free energy is given by



F  kTN ln Z  kTN ln 
 z

Vd
N
 G 
d
z
 d 
   .
 z 


From this expression we may now obtain all the
thermodynamic properties of the system, in particular
the internal energy is
U  NkT 2
 ln Z d
 NkT .
T
z
Substituting the number of particles N in terms of the partition
function (Z=N in our normalization) the internal energy can be
written as,
d 
V d   
d z
zd

z .
U
kT


d
z
N
z
z
G
 
Whereas the entropy is
 d  d z z
V d     k 
d
d
d

z
z


z
S
T
.


d
z
d
N
z
z
G
 
the relationship between energy and entropy in our
system is given by
STd
U
.
dz
This relationship is exactly the obtained in the case of
black branes.
Concluding remarks
We have introduced an action invariant under anisotropic
transformations in space and time.
This anisotropic mechanical system is consistent with the
non-relativistic Horava's dispersion relation.
Our system corresponds to the Conformal Mechanics for
z=2 and is a generalization of this system for arbitrary z.
We show that the full Schroedinger algebra constructed
from our generators does not close.
However we found that a subalgebra SL(2,R) indeed
close.
In the massless case the Horava particle is only a
nonlinear realization of the relativistic particle.
We remark that the thermodynamic properties of our
model reproduces the same thermodynamic properties
of the recent anisotropic black branes.