Field quantization via discrete approximations: problems and perspectives. Jerzy Kijowski Center for Theoretical Physics PAN Warsaw, Poland

2 QFTG - REGENSBURG Oct. 1, 2010 Perturbative QFT Conventional (perturbative) approach to quantum field theory: Reason (???) 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. : violation of locality principle.

3 QFTG - REGENSBURG Oct. 1, 2010 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. Lattice formulation of QFT Quantization of such an approximative theory is straightforward (up to minor technicalities) and leads to a model of quantum mechanical type.

4 QFTG - REGENSBURG Oct. 1, 2010 Hopes Lattice formulation of QFT 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. Weak version: There exists a limiting procedure which enables us to construct a coherent Quantum Field Theory as a limit of all these Quantum Mechanical systems. Strong version: Spacetime structure in micro scale???

5 QFTG - REGENSBURG Oct. 1, 2010 To define such a limit we must organize the family of all these discrete approximations of a given theory into an inductive-projective family. Lattice formulation of QFT Example: scalar (neutral) field. Typically (if the kinetic part of the Lagrangian function,,L’’ is quadratic), we have:. Field degrees of freedom described in the continuum version of the theory by two functions: – field Cauchy data on a given Cauchy surface.

6 QFTG - REGENSBURG Oct. 1, 2010 Scalar field For any pair and of vectors tangent to the phase space we have: Phase space of the system describes all possible Cauchy data: Symplectic structure of the phase space:

7 QFTG - REGENSBURG Oct. 1, 2010 Discretization – classical level 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 Choose a finite volume of the Cauchy surface and its finite covering (lattice) : Symplectic form of the continuum theory generates:

8 QFTG - REGENSBURG Oct. 1, 2010 There is a partial order in the set of discretizations: Hierarchy of discretizations Every lattice generates the Poisson algebra of classical observables, spanned by the family. Example 1:

9 QFTG - REGENSBURG Oct. 1, 2010 Hierarchy of discretizations Example 2:(intensive instead of extensive observables)

10 QFTG - REGENSBURG Oct. 1, 2010 Hierarchy of discretizations Example 2:(intensive instead of extensive observables)

11 QFTG - REGENSBURG Oct. 1, 2010 Hierarchy of discretizations Example 2:(intensive instead of extensive observables)

12 QFTG - REGENSBURG Oct. 1, 2010 Discretization – quantum level Quantum version of the system can be easily constructed on the level of every finite approximation. generate the quantum version of the observable algebra. Quantum operators: Schrödinger quantization: pure states described by wave functions form the Hilbert space. Different functional-analytic framework might be necessary to describe constraints (i.e. algebra of compact operators). Simplest version:

13 QFTG - REGENSBURG Oct. 1, 2010 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 ).

14 QFTG - REGENSBURG Oct. 1, 2010 Inductive system of quantum observables Example: Annihilator of generated by:

15 QFTG - REGENSBURG Oct. 1, 2010 Hierarchy of discretizations Example 2:(intensive observables) Anihilator of generated by:

16 QFTG - REGENSBURG Oct. 1, 2010 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!)

17 QFTG - REGENSBURG Oct. 1, 2010 Quantum states Quantum states (not necessarily pure states!) are functionals on the observable algebra. Projection mapping for states defined by duality: On each level of lattice approximation states are represented by positive operators with unital trace:

18 QFTG - REGENSBURG Oct. 1, 2010 Quantum states „Forgetting” about the remaining degrees of freedom. Physical state of the big system implies uniquely the state of its subsystem EPR

19 QFTG - REGENSBURG Oct. 1, 2010 Projective system of quantum states Chain rule satisfied: States on the complete algebra can be described by the projective limit:

20 QFTG - REGENSBURG Oct. 1, 2010 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).

21 QFTG - REGENSBURG Oct. 1, 2010 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.

22 QFTG - REGENSBURG Oct. 1, 2010 Special case: gauge and constraints Mixed (intensive-exstensive) representation of gauge fields: Implementation of constraints on quantum level: Gauge-invariant wave functions (if gauge orbits compact!), otherwise: representation of observable algebras. parallell transporter on every lattice link.

23 QFTG - REGENSBURG Oct. 1, 2010 General relativity theory Einstein theory of gravity can be formulated as a gauge theory. Field equations: After reduction: Lorentz group Possible discretization with gauge group:. Further reduction possible with respect to What remains?Boosts! Affine variational principle: first order Lagrangian function depending upon connection and its derivatives (curvature).

24 QFTG - REGENSBURG Oct. 1, 2010 General relativity theory This agrees with Hamiltonian formulation of general relativity in the complete, continuous version: Cauchy data on the three-surface : three-metric and the extrinsic curvature. Extrinsic curvature describes boost of the vector normal to when dragged parallelly along.

25 QFTG - REGENSBURG Oct. 1, 2010 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:

26 QFTG - REGENSBURG Oct. 1, 2010 Loop quantum gravity The theory is based on inductive system of quantum states! contains more links or gives finer description of the same links: But

27 QFTG - REGENSBURG Oct. 1, 2010 Inductive system of quantum states State of a subsystem determines state of the big system!!! contains more links or gives finer description of the same links: Inductive mapping of states: is a subsystem of the ``big system’’

28 QFTG - REGENSBURG Oct. 1, 2010 LQG - difficulties Lack of any reasonable approach to constraints. Leads to non-separable Hilbert spaces. State of a subsystem determines state of the big system!!! But the main difficulty is of physical (not mathematical) nature: Positivity of gravitational energy not implemented. Non-compact degrees of freedom excluded a priori.

29 QFTG - REGENSBURG Oct. 1, 2010 LQG - hopes State of a subsystem determines state of the big system!!! A new discrete approximation of the 3-geometry (both intrinsic and extrinsic!), compatible with the structure of constraints. Representation of the observable algebra on every level of discrete approximations. Positivity of gravitational energy implemented on every level of discrete approximations of geometry. Replacing of the inductive by a projective system of quantum states. State of a system determines state of the subsystem!!!

30 QFTG - REGENSBURG Oct. 1, 2010 References