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From Classical to Quantum Field Theories: Perturbative and Nonperturbative Aspects

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29 pages
Introduction Kinematical Structures Dynamical Structures Consequences Quantization and renormalization From Classical to Quantum Field Theories: Perturbative and Nonperturbative Aspects Romeo Brunetti Universita di Trento, Dipartimento di Matematica (Jointly with K. Fredenhagen (Hamburg), M. Dutsch (Gottingen) and P. L. Ribeiro (Sao-Paolo)) Lyon 16.VI.2010 Romeo Brunetti From classical to quantum field theories

  • dynamical structures

  • theory mainly via geometric techniques

  • regularity properties

  • structural foundations

  • universita di

  • quantum field


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Introduction
Kinematical Structures
Dynamical
Consequences
Quantization and renormalization
From Classical to Quantum Field Theories: Perturbative and
Nonperturbative Aspects
Romeo Brunetti
Universita di Trento, Dipartimento di Matematica
(Jointly with K. Fredenhagen (Hamburg), M. Dutsch (Gottingen) and P. L. Ribeiro
(Sao-Paolo))
Lyon 16.VI.2010
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures
Dynamical
Consequences
Quantization and renormalization
1 Introduction
2 Kinematical Structures
Support Properties
Regularity Properties
Results
3 Dynamical Structures
Lagrangians
Dynamics
M ller Scattering
Peierls brackets
4 Consequences
Structural consequences
Local covariance
5 Quantization and renormalization
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures
Dynamical
Consequences
Quantization and renormalization
Introduction
Typical (rigorous) approaches to Classical Field Theory mainly via geometric
techniques ((multi)symplectic geometry (Kijowski, Marsden et alt.), algebraic
geometry/topology (Vinogradov)) whereas physicists (B. de Witt) like to deal
with (formal) functional methods, tailored to the needs of (path-integral-based)
quantum eld theory. In this last case we have:
Heuristic in nite-dimensional generalisation of Lagrangian mechanics;
Making it rigorous is possible { usually done in Banach spaces
However, one of our results entails that:
Classical eld theory is not as \in nite dimensional" as it appears!
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures
Dynamical
Consequences
Quantization and renormalization
Aims/Bias
Structural Foundations: We wish to give a fresh look, along the algebraic
setting, of interacting classical eld theories. From that a \new"
quantization procedure for perturbation theory.
pAQFT: Many structures suggested by perturbation theory in the algebraic
fashion [Dutsch-F redenhagen (CMP-2003), Brunetti-Dutsch-F redenhagen
(ATMP-2009) , Brunetti-Fredenhagen (LNP-2009), Keller (JMP-2009)]
Setting
Model: Easiest example, real scalar eld ’
Geometry: The geometric arena is the following: (M; g) globally
hyperbolic Lorentzian manifold ( xed, but otherwise generic dimensionp
d 2), with volume element d = j det gjdxg
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures
Dynamical
Consequences
Quantization and renormalization
States and observables of classical eld theory
We mainly need to single out
STATES &OBSERVABLES
Reminder
In classical mechanics, states can be seen as points of a smooth nite
dimensional manifold M (con guration space) and observables are taken to be
1the smooth functions over it C (M). Moreover, we know that it has also a
Poisson structure. This is the structure we would like to have;
CONFIGURATION SPACE { OBSERVABLES! Kinematics
POISSON STRUCTURE! Dynamics
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures Support Properties
Dynamical Regularity Properties
Consequences Results
Quantization and renormalization
Con guration Space
We start with the
CONFIGURATION SPACE
1Also motivated by the nite-dimensional road map, we choose ’2 C (M; )
1with the usual Frechet topology (simpli ed notation E C (M; ))
This choice corresponds to what physicists call
OFF-SHELL SETTING
namely, we do not consider solutions of equations of motion (which haven’t yet
been considered at all!)
Romeo Brunetti From classical to quantum eld theoriesRRIntroduction
Kinematical Structures Support Properties
Dynamical Regularity Properties
Consequences Results
Quantization and renormalization
Observables
As far as observables are concerned, we de ne them (step-by-step)
F :E!
i.e. real-valued non-linear functionals.
The -linear space of all functionals is certainly an associative commutative
algebra F (M ) under the pointwise product de ned as00
(F:G)(’) = F (’)G(’)
However, in this generality not much can be said. We need to restrict the class
of functionals to have good working properties:
Restrictions
Support Properties
Regularity Properties
Romeo Brunetti From classical to quantum eld theoriesRRIntroduction
Kinematical Structures Support Properties
Dynamical Regularity Properties
Consequences Results
Quantization and renormalization
Support
De nition: Support
We de ne the spacetime support of a functional F as
:
suppF =Mnfx2M :9U3 x open s.t.8 ; ; supp U; F (+ ) = F ( )g
Lemma: Support properties
Usual properties for the support
Sum: supp(F + G) supp(F )[ supp(G)
Product: supp(F:G) supp(F )\ supp(G)
We require that all functionals have COMPACT support.
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures Support Properties
Dynamical Regularity Properties
Consequences Results
Quantization and renormalization
One further crucial requirement is
Additivity
If for all ; ; 2E such that supp \ supp =?, then1 2 3 1 3
F ( + + ) = F ( + ) F ( ) + F ( + );1 2 3 1 2 2 2 3
This replaces sheaf-like properties typical of distributions (in fact, it is a weak
replacement of linearity) and that allows to decompose them into small pieces.
Indeed,
Lemma
Any additive and compactly supported functional can be decomposed into
nite sums of such functionals with arbitrarily small supports
Additivity goes back to Kantorovich (1938-1939)!
Romeo Brunetti From classical to quantum eld theoriesIntroduction
Kinematical Structures Support Properties
Dynamical Regularity Properties
Consequences Results
Quantization and renormalization
Regularity
We would like to choose a subspace of the space of our functionals which
resembles that of the observables in classical mechanics, i.e. smooth
observables. We considerE our manifold but is not even Banach, so one needs
a careful de nition of di erentiability
[Michal (PNAS-USA-1938!), Bastiani (JAM-1964), popularized by Milnor (Les
Houches-1984) and Hamilton (BAMS-1982)]
De nition
The derivative of a functional F at ’ w.r.t. the direction is de