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Field quantization via discrete
approximations:
problems and perspectives.
Jerzy Kijowski
Center for Theoretical Physics PAN
Warsaw, Poland
Perturbative QFT
Conventional (perturbative) approach to quantum field theory:
1) Fundamental objects are plane waves.
2) Their (non-linear) interaction is highly non-local.
In spite of almost 80 years of (unprecedented) successes,
this approach has probably attained the limits of its
applicability.
Reason (???) : violation of locality principle.
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Lattice formulation of QFT
Alternative (non-perturbative) approach, based on
discrete approximations:
1) Physical system with infinite number of degrees of freedom
(field) replaced by a system with finite number of (collective)
degrees of freedom (its lattice approximations).
2) These collective degrees of freedom are local.
They interact according to local laws.
Quantization of such an approximative theory is straightforward
(up to minor technicalities) and leads to a model of quantum
mechanical type.
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Lattice formulation of QFT
Hopes
Weak version:
Quantizing sufficienly many (but always finite number) of
degrees of freedom we will obtain a sufficiently good
approximation of Quantum Field Theory by
Quantum Mechanics.
Strong version:
Spacetime structure
in micro scale???
There exists a limiting procedure which enables us to
construct a coherent Quantum Field Theory as a limit
of all these Quantum Mechanical systems.
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Lattice formulation of QFT
To define such a limit we must organize the family of all these
discrete approximations of a given theory into an
inductive-projective family.
Example: scalar (neutral) field.
Field degrees of freedom described in the continuum version
of the theory by two functions:
– field Cauchy data
on a given Cauchy surface
.
Typically (if the kinetic part of the Lagrangian function ,,L’’
is quadratic), we have:
.
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Scalar field
Phase space of the system describes all possible Cauchy data:
Symplectic structure of the phase space:
For any pair
and
vectors tangent to the phase space we have:
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of
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Discretization – classical level
Choose a finite volume
of the Cauchy surface
and its finite covering (lattice)
:
where
are finite (relatively compact) and ``almost disjoint’’
(i.e. intersection
has measure zero for
).
Define the finite-dimensional algebra of local observables:
Possible manipulations
of the volume factors
Symplectic form of the continuum theory generates:
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Hierarchy of discretizations
Every lattice
generates the Poisson algebra
of classical observables, spanned by the family
.
There is a partial order in the set of discretizations:
Example 1:
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Hierarchy of discretizations
Example 2: (intensive instead of extensive observables)
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Hierarchy of discretizations
Example 2: (intensive instead of extensive observables)
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Hierarchy of discretizations
Example 2: (intensive instead of extensive observables)
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Discretization – quantum level
Quantum version of the system can be easily constructed
on the level of every finite approximation .
Schrödinger quantization: pure states described by wave
functions
form the Hilbert space
.
Quantum operators:
generate the quantum version
of the observable algebra.
Simplest version:
Different functional-analytic framework might be necessary
to describe constraints (i.e. algebra of compact operators).
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Inductive system of quantum
observables
Observable algebras
form an inductive system:
Theorem:
Definition:
More precisely: there is a natural embedding
.
Proof:
where
describes ``remaining’’ degrees of freedom
(defined as the symplectic annihilator of
).
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Inductive system of quantum
observables
Example:
Annihilator
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of
generated by:
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Hierarchy of discretizations
Example 2: (intensive observables)
Anihilator
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of
generated by:
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Inductive system of quantum
observables
Embeddings
are norm-preserving and satisfy the chain rule:
Complete observable algebra can be defined as the inductive
limit of the above algebras, constructed on every level of the
lattice approximation:
(Abstract algebra. No Hilbert space!)
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Quantum states
Quantum states (not necessarily pure states!) are functionals
on the observable algebra.
On each level of lattice approximation states are represented
by positive operators with unital trace:
Projection mapping for states defined by duality:
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Quantum states
Physical state of the big system
its subsystem
implies uniquely the state of
„Forgetting” about the remaining degrees of freedom.
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EPR
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Projective system of quantum states
Chain rule satisfied:
States on the complete algebra
by the projective limit:
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can be described
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Hilbert space
Given a state
on the total observable algebra
(a ``vacuum state’’), one can generate the appropriate
QFT sector (Hilbert space) by the GNS construction:
Hope:
The following construction shall (maybe???) lead to the
construction of a resonable vacuum state:
1) Choose a reasonable Hamilton operator
on every level
of lattice approximation (replacing derivatives by differences
and integrals by sums).
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Hilbert space
Hope:
The following construction shall (maybe???) lead to the
construction of a resonable vacuum state:
1) Choose a reasonable Hamilton operator
on every level
of lattice approximation (replacing derivatives by differences
and integrals by sums).
2) Find the ground state
of
.
3) (Hopefully) the following limit does exist:
Corollary:
is the vacuum state of the complete theory.
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Special case: gauge and constraints
Mixed (intensive-exstensive) representation of gauge fields:
parallell transporter
on every lattice link .
Implementation of constraints on quantum level:
Gauge-invariant wave functions (if gauge orbits compact!),
otherwise: representation of observable algebras.
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General relativity theory
Einstein theory of gravity can be formulated as a gauge theory.
Affine variational principle: first order Lagrangian function
depending upon connection and its derivatives (curvature).
Field equations:
Possible discretization with gauge group:
.
After reduction: Lorentz group
Further reduction possible with respect to
What remains?
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Boosts!
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General relativity theory
This agrees with Hamiltonian formulation of general relativity
in the complete, continuous version:
Cauchy data on the three-surface
extrinsic curvature
.
: three-metric
Extrinsic curvature describes boost of the vector
when dragged parallelly along .
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and the
normal to
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Loop quantum gravity
The best existing attempt to deal with quantum aspects of gravity!
In the present formulation: a lattice gauge theory with
.
Why?
But:
After reduction with respect to rotations we end up with:
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Loop quantum gravity
contains more links or gives finer description of the same links:
But
The theory is based on inductive system of quantum states!
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Inductive system of quantum states
contains more links or gives finer description of the same links:
is a subsystem of the ``big system’’
Inductive mapping of states:
State of a subsystem determines state of the big system!!!
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LQG - difficulties
Leads to non-separable Hilbert spaces.
Positivity of gravitational energy not implemented.
Lack of any reasonable approach to constraints.
Non-compact degrees of freedom excluded a priori.
But the main difficulty is of physical (not mathematical) nature:
State of a subsystem determines state of the big system!!!
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LQG - hopes
A new discrete approximation of the 3-geometry (both intrinsic
and extrinsic!), compatible with the structure of constraints.
Positivity of gravitational energy implemented on every level
of discrete approximations of geometry.
Representation of the observable algebra on every level of
discrete approximations.
Replacing of the inductive by a projective system of quantum
states.
State of a subsystem determines state of the big system!!!
State of a system determines state of the subsystem!!!
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References
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