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Caricatures of Big Bang from Matrices

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Presented By:
Sumit R. Das
University of Kentucky, Lexington

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Space-time from Matrices
A common slogan in string theory is that space and time are not fundamental, but derived concepts which emerge out of more fundamental structures.
In a few cases we have some hint of what this structure could be “ these are situations where the space-time physics has a holographic description “ usually in terms of a field theory of matrices.
These are in fact description of closed string dynamics in terms of open strings
Examples
Closed String Theory Open String Theory
2 dimensional strings Matrix Quantum Mechanics
M theory/ critical string SUSY Matrix Quantum
in light cone gauge Mechanics/ 1+1 YM
Strings in 3+1 dimensional N=4
Yang-Mills
These holographic descriptions have played a crucial role in our understanding of puzzling aspects of quantum gravitational physics, e.g. Black Holes
Can we use these to address some puzzling questions in time-dependent space-times “ in particular cosmologies where time appears to begin or end “ e.g. Big Bangs or Big crunches ?
Can we ask what do we even mean when we say that time begins or ends ?
In this talk I will discuss some recent attempts in this direction.
2d Closed String from Double scaled Matrix Quantum Mechanics
- hermitian matrix. This is the degree of freedom of open strings joining D0 branes
Gauging “ states are singlet under SU(N)
Eigenvalues are fermions. Single particle hamiltonian
Density of fermions
To leading order in 1/N, the dynamics of the scalar field is given by the action
This collective field theory would be in fact the field theory of closed strings in two dimensions “ the space dimension has emerged out of the matrix
The fundamental quantum description is in terms of fermions
Collective field theory used to find the emergent space-time as seen by closed strings
Ground State and fluctuations
Filled fermi sea
Collective field
Fluctuations
Two scalar fields for the two sides.
The quadratic action for fluctuations
Metric determined up to conformal factor
There are actually two fields for the two branches of the fermi surface
Space time structure is transparent in Minkowskian coordinates
is independent of time
Any other conformal frame will destroy this property
A Time-dependent solution
S.R.D. and J. Karczmarek, PRD D71 (2005) 086006
An infinite symmetry of the theory generates time dependent solutions.
One example
The space-time generated has a space like boundary
Similarly a time-reversed solution
This is a geodesically incomplete space-time. The space-like boundary has regions of strong coupling
Normally one would simply extend the space-time to complete it
However in this case there is a fundamental definition of time provided by the matrix model “ t
The space-time perceived by closed strings is an emergent description
Lesson
The open string time can go over the full range
The closed string time can be terminated
The underlying theory of open strings is still that of free fermions “ but there is no clear space-time interpretation.
Space-time in Matrix String Theory for Type IIA
By a standard chain of dualities, Type IIA string theory with a compact light cone direction with radius R and with momentum
is equivalent to 1+1 dimensional SU(J) Yang-Mills theory with
on a spatial circle of radius
Time dependent couplings
Craps, Sethi and E.Verlinde : Matrix String Theory in a background with flat string frame metric, but a dilaton
An alternative view
Equivalently the YM theory can be thought to have a constant coupling, but living on the future wedge of the Milne universe
PP Wave Big Bangs
S.R.D. and J. Michelson-Phys.Rev.D72(2005)086005,
S.R.D, J. Michelson,¦¦¦ to appear
Motivation : to find a situation where there is some control of the precise nature of non-abelian excitations
Possible for pp-wave solutions with null linear dilatons
For example in IIB theory
IIB closed string on a 2-torus with some momentum along one direction
This is dual to a 2+1 dimensional YM theory on a 2-torus
Matrix membrane for usual pp-waves
For time-independent pp-waves (Q=0) and for weak IIB coupling
(i) only diagonal Xâ„¢s survive
(ii) The gauge field gets dualized into a scalar field “ so we have 8 scalars now
(iii) The size of the direction is small “ effectively becomes a 1+1 dimensional theory
(iv) This 1+1 dimensional theory becomes the world-sheet theory of the original IIB string moving in this background
Fuzzy ellipsoids
: semiclassical limit in which classical solutions important
In this case the classical solutions are fuzzy ellipsoids formed by Myersâ„¢ effect with time-dependent sizes
Time evolution
For generic initial conditions at the big bang the size of the fuzzy ellipsoids vanish at late times
Similarly, very small fuzzy ellipsoids at late times grow large at early times
The time dependence of is responsible for releasing non-abelian degrees of freedom near the big bang
The time dependence in front of means that the masses of the Kaluza Klein modes in the direction is time dependent
In terms of the original IIB theory these KK modes are D1 branes wrapped around
Particle depletion
This results in production or depletion of these KK modes with time
For scalars, the is defined in terms of the modes
The vacuum is defined in terms of the modes
Correspondingly there are creation operators
and which are not equivalent
The nontrivial Bogoliubov relations imply that
If the state at late time does not contain any of these modes, the state at early time contains lots of pairs of these particles in a squeezed state
Big Bangs in AdS/CFT
The time independent IIB pp-wave has another holographic dual “ this is the large R charge sector of a 3+1 dimensional N=4 YM theory
Can we extend this duality to our null dilaton pp-wave ?
This requires a deformation of geometry whose Penrose limit is this pp-wave.
We haveâ„¢nt found this yet
If we succeed, we can pose interesting cosmological questions in this gauge theory.
Conclusions
In the toy models considered in this talk, what appears to be initial or final singularities from the point of view of closed string theory (and hence usual gravity) are not really singular in the holographic theory
Rather in these regions the reduction of degrees of freedom which leads to an interpretation of the space of matrices as space-time is not valid “ in some cases the full noncommutative nature of the theory becomes significant
Mapping these problems into problems in gauge theory is likely to yield significant insight