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Field quantization via discrete approximations: problems and perspectives.

Field quantization via discrete approximations: problems and perspectives. . Jerzy Kijowski Center for Theoretical Physics PAN Warsaw, Poland. 1) Fundamental objects are plane waves . . 2) Their (non-linear) interaction is highly non-local . Perturbative QFT.

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Field quantization via discrete approximations: problems and perspectives.

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

  2. 1) Fundamental objects are plane waves. 2) Their (non-linear) interaction is highly non-local. Perturbative QFT Conventional (perturbative) approach to quantum field theory: In spite of almost 80 years of (unprecedented) successes, this approach has probably attained the limits of its applicability. Reason (???) : violation of locality principle. QFTG - REGENSBURG

  3. 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). Lattice formulation of QFT Alternative (non-perturbative) approach, based on discrete 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. QFTG - REGENSBURG

  4. 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. Spacetime structure in micro scale??? Strong 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. Lattice formulation of QFT Hopes QFTG - REGENSBURG

  5. 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: . 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. QFTG - REGENSBURG

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

  7. 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 generates: Symplectic form of the continuum theory Discretization – classical level QFTG - REGENSBURG

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

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

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

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

  12. 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). Discretization – quantum level QFTG - REGENSBURG

  13. 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 ). Inductive system of quantum observables QFTG - REGENSBURG

  14. Example: Annihilator of generated by: Inductive system of quantum observables QFTG - REGENSBURG

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

  16. 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: Inductive system of quantum observables (Abstract algebra. No Hilbert space!) QFTG - REGENSBURG

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

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

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

  20. 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: • Choose a reasonable Hamilton operator on every level • of lattice approximation (replacing derivatives by differences • and integrals by sums). Hilbert space QFTG - REGENSBURG

  21. Hope: The following construction shall (maybe???) lead to the construction of a resonable vacuum state: 2) Find the ground state of . 3) (Hopefully) the following limit does exist: • Choose a reasonable Hamilton operator on every level • of lattice approximation (replacing derivatives by differences • and integrals by sums). Corollary: Hilbert space is the vacuum state of the complete theory. QFTG - REGENSBURG

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

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

  24. 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 . General relativity theory This agrees with Hamiltonian formulation of general relativity in the complete, continuous version: QFTG - REGENSBURG

  25. In the present formulation: a lattice gauge theory with . But: After reduction with respect to rotations we end up with: Loop quantum gravity The best existing attempt to deal with quantum aspects of gravity! Why? QFTG - REGENSBURG

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

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

  28. But the main difficulty is of physical (not mathematical) nature: State of a subsystem determines state of the big system!!! 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. QFTG - REGENSBURG

  29. State of a system determines state of the subsystem!!! State of a subsystem determines state of the big system!!! 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. QFTG - REGENSBURG

  30. References QFTG - REGENSBURG

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