THE “EXTENDED PHASE SPACE” APPROACH
TO QUANTUM GEOMETRODYNAMICS:
WHAT CAN IT GIVE FOR THE DEVELOPMENT
OF QUANTUM GRAVITY
T. P. Shestakova
Department of Theoretical and Computational Physics
Southern Federal University
(former Rostov State University)
Sorge St. 5, Rostov-on-Don 344090, Russia
E-mail: shestakova@phys.rsu.ru
The “extended phase space” approach
to Quantum Geometrodynamics
G. M. Vereshkov, V. A. Savchenko and T. P. Shestakova,
Rostov State University, the end of 1990s.

V. A. Savchenko, T. P. Shestakova, G. M. Vereshkov, "Quantum
Geometrodynamics of the Bianchi IX model in extended phase
space", Int. J. Mod. Phys. A14 (1999), P. 4473 – 4490.

V. A. Savchenko, T. P. Shestakova, G. M. Vereshkov, "The exact
cosmological solution to the dynamical equations for the Bianchi
IX model", Int. J. Mod. Phys. A15 (2000), P. 3207 – 3220.

V. A. Savchenko, T. P. Shestakova, G. M. Vereshkov, "Quantum
Geometrodynamics in extended phase space - I. Physical
problems of interpretation and mathematical problems of gauge
invariance", Gravitation & Cosmology 7 (2001), P. 18 – 28.

V. A. Savchenko, T. P. Shestakova, G. M. Vereshkov, "Quantum
Geometrodynamics in extended phase space - II. The Bianchi IX
model", Gravitation & Cosmology 7 (2001), P. 102 - 116
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The criticism of the Wheeler – DeWitt theory
“...although it may be heretical to suggest it, the Wheeler –
DeWitt equation – elegant though it be –may be completely
the wrong way of formulating a quantum theory of
gravity''.
C. Isham,
“Canonical quantum gravity and the problem of time”,
Preprint gr-qc/9210011.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The Wheeler – DeWitt Quantum Geometrodynamics:
the first significant attempt to construct full
quantum theory of gravity
The three cornerstones on which the Wheeler – DeWitt quantum
geometrodynamics is based:
•
Dirac approach to quantization of systems with constraints
•
Arnowitt – Deser – Misner (ADM) parametrization
•
The ideas of Wheeler concerning a wave functional describing a
state of gravitational field
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The role of constraints: Dirac postulate
Each constraint
m  q, p   0
after quantization becomes a condition on a state vector, or wave
functional
m   0
This is a postulate which cannot be justified by the reference to the
correspondence principle.
At the classical level, the constraints express gauge invariance of the
theory.
What grounds do we have to expect that the same takes place
at the quantum level?
Could we consider quantum geometrodynamics as a gaugeinvariant theory?
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The role of ADM parametrization:
constraints can be written in the form
independent of gauge variables
ADM parametrization has a clear geometrical interpretation.
Gravitational constraints
H  0; H i  0


H  g K ij K ij  K 2  3 R  Gijkl ij kl  g 3 R
1
Gijkl 
g  gik g jl  gil g k  gij g kl 
2
H i  2 ;ijj  2 j ij  g il  2 k g jl   l g jk   jk
do not depend on the lapse and shift functions N , Ni
However, it does not ensure that quantum theory based on
constraint equations would be gauge invariant
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The role of ADM parametrization:
constraints can be written in the form
independent of gauge variables
At the same time, the ADM parametrization introduces (3+1)splitting in 4-dimensional spacetime.
It is equivalent to a choice of a reference frame, and gauge
invariance breaks down.
Thus, the Hamiltonian constraint loses its sense and, with the
latter, so does the whole procedure of quantization.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The ideas of Wheeler concerning a wave functional
describing a state of gravitational field
The wave functional must be determined on the superspace of all
possible 3-geometries and depend only on 3-geometry.
This statement remains to be just a declaration without any
mathematical realization. The state vector always depends on
a concrete form of the metric.
    g  
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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Path integral approach vs. canonical approach
•
•
Canonical approach: no strict proof of gauge invariance of the
Wheeler − DeWitt theory
Path integral approach contains the procedure of derivation of
an equation for a wave function from the path integral, while
gauge invariance of the path integral, and the theory as a whole,
being ensured by asymptotic boundary conditions.
In ordinary quantum theory one usually considers systems with
asymptotic states in which the so-called physical and non-physical
degrees of freedom could be separated from each other.
Asymptotic boundary conditions in the path integral are equivalent
to selection rules for physical states.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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Gauge invariance of Quantum Geometrodynamics
Path integral approach: problem of asymptotic states
In first works devoted to derivation of the Wheeler − DeWitt
equation from the path integral asymptotic boundary conditions
were tacitly adopted
A. O. Barvinsky and V. N. Ponomariov “Canonical
Quantization of Gravity and Quantum Geometrodynamics” Phys.
Lett. B167 (1986), P. 289.
J. J. Halliwell “Derivation of the Wheeler - DeWitt equation
from a path integral for minisuperspace models” Phys. Rev. D38
(1988) P.2468 - 2481.
A Universe with non-trivial topology does not possess
asymptotic states!
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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Gauge invariance of Quantum Geometrodynamics
We are not sure that in the absence of asymptotic states in a
topologically non-trivial universe we would be able to construct a
gauge-invariant theory.
We have no grounds at all to require for a wave function to satisfy
the Wheeler − DeWitt equation.
At the same time, independently on our notion about gauge
invariance or noninvariance of the theory, the wave function has
to obey some Schrödinger equation.
Only after constructing the wave function satisfying the Schrödinger
equation, we shall be able to investigate the question, if this wave
function obey the Wheeler − DeWitt equation as well.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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Parametrization noninvariance
The choice of gauge variables and the choice of gauge conditions
have a unified interpretation: they together determine equations
for the metric components, fixing a reference frame
Parametrization

g 0   v N ,  ij

+ Gauge conditions
N  f   ij 

Equation for

g0 :
g0   v f  ij  ,  ij

To define the operator form of the constraints after quantization, we
do need to know the relations between the gauge variables and
the rest variables.
S. W. Hawking, D. N. Page, "Operator ordering and the
flatness of the Universe", Nucl. Phys. B264 (1986), P. 185.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The fundamental problems of the Wheeler − DeWitt theory







The
The
The
The
The
The
…..
problem of time (the absence of dynamical evolution)
problem of Hilbert space
problem of observables
problem of reparametrization noninvariance
ordering problem
problem of global structure of spacetime
All these problems are interrelated and cannot be solved in
the frameworks of the Wheeler − DeWitt theory.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The choice of quantization scheme


The path integral in Lagrangian form with Batalin − Vilkovisky
(Faddeev − Popov) effective action.
The path integral in Hamiltonian form with Batalin − Fradkin −
Vilkovysky effective action with following integrating out all
momenta and passing on to a path integral over extended
configurational space.

The difference between the group of transformations
generated by gravitational constrains in Hamiltonian formalism
and that of gauge transformations of the Einstein theory (in
Lagrangian formalism).
The two formulations could enter into agreement only in a
gauge-invariant sector which can be singled out by asymptotic
boundary conditions
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The Universe is a system
which, in general, does not possess asymptotic states
We have come to the following questions:



What formalism should one prefer?
What are consequences of the fact that we consider the path
integral without asymptotic boundary conditions?
What will be a role of gauge degrees of freedom, which were
traditionally considered as redundant, in this new approach?
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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A simple cosmological model
with a finite number of degrees of freedom
The action:
 1

1
a b
S   dt  v N , Q  abQ Q 
U  Q    0 N  f aQ  iw N , Q  
v N, Q
 2







The arbitrary parametrization of a gauge variable:


w N, Q 

v N, Q
v, N
;
def
v, N 




a3
 v N, Q
N
v
N
N f
The class of gauge conditions
can be presented in a differential form
Q  k;
k  const.
N  f,aQ ;
a
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
f
f,a 
.
a
Q
def
16
Hamiltonian dynamics in extended phase space
Though we do not need the Hamiltonian formulation of the theory to
derive a Schrödinger equation from the path integral, the
differential form of gauge conditions enables us to construct the
Hamiltonian in a usual way
1 
1
i
H  PaQ   0 N      L  G P P 
U Q  

2
v N, Q
w N, Q
a

   0, a  ; Q0  N ; G

 f,a f ,a
1

 ,a
v N, Q  f


T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …


f ,a 
.
ab 
 
17
The equivalence between
Lagrangian and Hamiltonian formulations
Varying the effective action with respect to physical variables Q ,
the gauge variable N , the Lagrange multiplier  0 and ghosts  ,
one gets, correspondingly, motion equations for physical
variables, the constraint, the gauge condition and equations for
ghosts. The extended set of Lagrangian equations is complete in
the sense that it enables one to formulate the Cauchy problem.
The set of Hamiltonian equations in extended phase space
H
H
P
; Q
Q
P
is completely equivalent to the extended set of Lagrangian
equations, the constraint and the gauge condition acquiring the
status of Hamiltonian equations. Gauge and ghost degrees of
freedom are treated on an equal basis with other variables.
This gave rise to Quantum Geometrodynamics in Extended
Phase Space.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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
The existence of two BRST generators
The group of transformations in extended phase space
corresponds to the group of gauge transformations in the
Lagrangian formalism. One can construct the BRST generator


  w N , Q  0  H  i 0   H
It generates transformations in extended phase space which are
identical to the BRST transformations in the Lagrangian formalism
In the Batalin − Fradkin − Vilkovisky approach we have the set of
constraints
G =  0 , T

and the BRST generator
 BFV   G  T  i 0 
BFV   0  T   0
  0  ?
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The Schrödinger equation
for the wave function of the Universe
The wave function is defined on extended configurational space with
the coordinates N , Q,  ,  .
i

 N , Q,  ,  ; t
t
  H
 N , Q,  ,  ; t 
i  
1 

1

H 

MG

U  V  f 



w   2M Q
Q
v


M N, Q  v
K
2
 N , Q w  N , Q
1
5
1
2
,a
,a
,a
w
f
f

2
w
f
w

w
w

w,  ,  f ,a f ,a  2w,  ,a f ,a  w,  f ,a,a  w,,aa  



, ,a
, ,a
,a
2
3w
12w
K 2
K 2  7K  6 2
,a
,a
,a
,a
,a
,a
,a

v,  w,  f,a f  v,  f,a w  w,  f ,a v  v,a w  
v
f
f

2
v
f
v

v
v




,

,
a
,

,
a
,
a
2
6vw
24v
1 K

v,  ,  f,a f ,a  2v,  ,a f ,a  v,  f,a,a  v,,aa  .

6v
Vf
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The general solution to the Schrödinger equation
The general solution




 N , Q,  ,  ; t    k  Q, t   N  f  Q   k   i  dk
is a superposition of eigenstates of a gauge operator
 N  f Q  k
k k ;


k   N  f Q   k .
It can be interpreted in the spirit of Everett's “relative state”
formulation. In fact, each element of the superposition describe a
state in which the only gauge degree of freedom N is definite,
so that time scale is determined by processes in the physical
subsystem.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The description of the physical subsystem
The function  k  Q, t  describes a state of the physical subsystem
for a reference frame fixed by the gauge condition
N  f  Q   k ; k  const.
It is a solution to the equation
i
 k  Q, t 
 H  phys   f   k  Q, t 
t
 1  1


1
ab
H  phys   f    
M
 U  V  
a
b
v
Q
 2M Q v
 N  f Q  k
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The normalization condition for the wave function
For the superposition defined above, the normalization condition
reads

 

*

  N , Q,  ,  ; t   N , Q ',  ,  ; t M N , Q dN d d

 
 
a
dQ
 
a

   *k  Q, t   k '  Q, t   N  f  Q   k  N  f  Q   k ' M N , Q dk dk ' dN
   *k  Q, t   k '  Q, t  M  f  Q   k , Q  dk
 dQ
a
a
dQ


a
 1.
a
The measure in physical subspace also depends on the gauge
condition!
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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What is the gauge subsystem?
The set of equations includes ghosts equations and a gauge
condition, and equations for physical degrees of freedom also
contain gauge noninvariant terms:

1
R    R   T mat   T obs   T ghost 
2

The Hamiltonian constraint can be presented in the form
H  E , E     gT00 obs  d 3 x
In quantum theory the modified Hamiltonian constraint leads to a
stationary Schrödinger equation for the physical part of the wave
function:
H  phys   kn  Q   En  kn  Q  ,  k  Q, t    cn  kn  Q  exp  iEn t 
n
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The isotropic universe: an example
The parametrization function as well as the gauge depend only on a
scale factor:


a3
 v N, a , N  f a  k
N
The quasi-energy-momentum tensor of the gauge subsystem reads:

T obs   diag   obs  ,  p obs  ,  p obs  ,  p obs 
  obs 
p obs 
2
0 v  N, a
 2
2
a 6 v, 

N  f  a  k


a
   obs  1 
v,  f,a  v,a  

 3v N , a

N  f  a  k


T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The gauge subsystem as a medium filling the Universe
The gauge-fixing term in the action describes a medium with the
equation of state depending on the chosen parametrization and
gauge.
If the gauge variable is the lapse function N , and the gauge
condition is
1
N a 3
a
the equation of the medium would be
At
At
a 0
a 
the equation gives
it gives
p obs 
p obs 
1 a4  3

  obs 
4
3 a 1
p obs    obs 
1
   obs 
3
We can see that the gauge subsystem appears to be a factor of
cosmological evolution; its state changing over the history of
the Universe determining a cosmological scenario.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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The description of the Universe
depends on a chosen reference frame
Gauge coordinate transformations

Changing of gauge condition fixing a reference frame

A new form of the Schrödinger equation
for a wave function of the Universe

Changing of solution to the Schrödinger equation

A new picture of the observable Universe
corresponding to a given reference frame
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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New problems




What is the Hilbert space structure for different reference frames?
What could be relations between the solutions to the Schrödinger
equation corresponding to various reference frames?
A possible mathematical task: to find classes of solutions within
which the solutions are “stable” enough under small variations of
gauge conditions. The structure of these classes must be anyhow
related with the structure of diffeomorphism group.
An alternative way: seeking for any “privileged” reference frame
in which the picture of the Universe evolution would better
correspond with observational data.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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New problems


Thermodynamical properties of a quantum Universe filled with a
medium playing the role of a reference frame.
The transition to classical limit of General Relativity when all
gauge-dependent effects must vanish.
We need for some mechanism which would explain how in the
result of quantum evolution the Universe appears to be in the
state with zero eigenvalue of the Hamiltonian. This mechanism
should be general enough not to depend on a chosen model.
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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Thank you
for your attention
T. P. Shestakova. The “extended phase space” approach to Quantum Geometrodynamics …
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