Precanonical quantization: from foundations to quantum gravity Igor - PowerPoint PPT Presentation

Precanonical quantization: from foundations to quantum gravity Igor Kanatchikov National Center of Quantum Information in Gda´ nsk (KCIK) Sopot, Poland E-mail: kanattsi@gmail.com, kai@fuw.edu.pl 32nd Workshop on Foundations and Constructive Aspects of QFT Wuppertal, Germany, May 31-June 1, 2013

Introduction and motivation Different strategies towards quantum gravity: • Apply QFT to GR (e.g. WDW, path integral) • Adapt GR to QFT (e.g. Ashtekar variables, Shape dynamics) • Change the fundamental microscopic dynamics, GR as an ef- fective emergent theory (e.g. string theory, GFT, induced grav- ity, quantum/non-commutative space-times)

Introduction and motivation • To try: Adapt quantum theory to GR? – Take the relativistic space-time seriously, – Avoid the distinguished role of time dimension in the for- malism of quantum theory. ! The distinguished role of time is rooted already in the classical canonical Hamiltonian formalism underlying canonical quantization. ! Fields as infinite-dimensional Hamiltonian systems evolving in time. How to circumvent it? ! De Donder-Weyl (”precanonical”) Hamiltonian formalism. Precanonical means that the mathematical structures of the canonical Hamiltonian formal- ism can be derived from those of the DW (or multisymplectic, or polysymplectic) formalism. Precanonical and canonical coincide at n = 1 . ! ”Precanonical quantization” is based on the structures of DW Hamiltonian formalism.

Outline of the talk • De Donder-Weyl Hamiltonian formulation. • Mathematical structures of DW theory. Poisson-Gerstenhaber brackets on forms. • Applications of P-G brackets: field equations, geometric prequantization. • Precanonical quantization of scalar field theory. • Precanonical quantization vs. functional Schr¨ odinger represen- tation. • Precanonical quantization of gravity. A. in metric variables, B. in vielbein variables. • Discussion.

De Donder-Weyl (precanonical) Hamiltonian formalism • Lagrangian density: L = L ( y a , y a µ , x ⌫ ) . • polymomenta: p µ a := @ L/ @ y a µ . • DW (covariant) Hamiltonian function: H := y a µ p µ a � L , ! H = H ( y a , p µ a , x µ ) . , • DW covariant Hamiltonian form of field equations: @ µ y a ( x ) = @ H/ @ p µ a , @ µ p µ a ( x ) = � @ H/ @ y a . � �� �� � � @ 2 L/ @ y µ � 6 = 0 . a @ y ⌫ • New regularity condition: det b ! No usual constraints, , ! No space-time decomposition, , ! Finite-dimensional covariant analogue of the , configuration space: ( y a , x µ ) .

De Donder-Weyl (precanonical) Hamiltonian formalism 2. DWHJ • DW Hamilton-Jacobi equation on n functions S µ = S µ ( y a , x ⌫ ) : ✓ ◆ a = @ S µ @ µ S µ + H y a , p µ @ y a , x ⌫ = 0 . • Can DWHJ be a quasiclassical limit of some Schr¨ odinger like formulation of QFT? • How to quantize fields using the DW analogue of the Hamiltonian formalism? The potential advantages would be: – Explicit compliance with the relativistic covariance principles, – Finite dimensional covariant analogue of the configuration space: ( y a , x µ ) instead of ( y ( x ) , t ) . • What are the Poisson brackets in DW theory? What is the ana- logue of canonically conjugate variables, the starting point of quantization? 6

DW Hamiltonian formulation: Examples Nonlinear scalar field theory L = 1 2 @ µ y @ µ y � V ( y ) DW Legendre transformation: p µ = @ L @@ µ y = @ µ y H = @ µ p µ � L = 1 2 p µ p µ + V ( y ) DW Hamiltonian equations: @ µ y ( x ) = @ H/ @ p µ = p µ , @ µ p µ ( x ) = � @ H/ @ y = � @ V/ @ y are equivalent to : 2 y + @ V/ @ y = 0 .

DW Hamiltonian formulation: Einstein’s gravity Einstein’s gravity in metric variables • ΓΓ action: L ( g ↵� , @ ⌫ g ↵� ); • Field variables: h ↵� := p g g ↵� , with g := | det( g µ ⌫ ) | , 1 • Polymomenta: Q ↵ 8 ⇡ G ( � ↵ ( � Γ � � ) � � Γ ↵ �� := �� ); • DW Hamiltonian density: H ( h ↵� , Q ↵ �� ) , ✓ ◆ H = p g H = 8 ⇡ G h ↵� 1 ↵� Q � 1 � n Q � Q � ↵� Q � �� + ; �� • Einstein field equations in DW Hamiltonian form @ ↵ h �� = @ H / @ Q ↵ �� , @ ↵ Q ↵ �� = � @ H / @ h �� . • No constraints analysis! – Gauge fixing is still necessary to single out physical modes.

Mathematical structures of the DW formalism. Brief outline. 1. Finite dimensional ”polymomentum phase space” ( y a , p µ a , x ⌫ ) 2. Polysymplectic ( n + 1) -form: Ω = dy a ^ dp µ a ^ ! µ , with ! µ = @ µ ( dx 1 ^ dx 2 ^ ... ^ dx n ) . 3. Horizontal differential forms F ⌫ 1 ... ⌫ p ( y, p, x ) dx ⌫ 1 ^ ... ^ dx ⌫ p as dynamical variables. 4. Poisson brackets on differential forms follow from X F Ω = dF . ) Hamiltonian forms F , ) Co-exterior product of Hamiltonian forms: p q p q F := ⇤ � 1 ( ⇤ F • F ^ ⇤ F ) . ) Graded Lie (Nijenhuis) bracket. ) Gerstenhaber algebra. 5. The bracket with H generates d • on forms. 6. Canonically conjugate variables from the analogue of the Heisenberg subalgebra: [ p µ a ! µ , y b ] } = � b [ p µ a ! µ , y b ! ⌫ ] } = � b [ p µ a , y b ! ⌫ ] } = � b a � µ { a , { a ! ⌫ , { ⌫ .

Geometric setting 1 • Classical fields y a = y a ( x ) are sections in the covariant config- uration bundle Y ! X over an oriented n -dimensional space- time manifold X with the volume form ! . • local coordinates in Y ! X : ( y a , x µ ) . • V p q ( Y ) denotes the space of p -forms on Y which are annihilated by ( q + 1) arbitrary vertical vectors of Y . • V n 1 ( Y ) ! Y : - generalizes the cotangent bundle, - models the multisymplectic phase space. • Multisymplectic structure: a dy a ^ ! µ + p ! , Θ MS = p µ ! µ := @ µ ! . • A section p = � H ( y a , p µ a , x ⌫ ) yields the Hamiltonian Poincar´ e- Cartan form Ω PC : a ^ dy a ^ ! µ + dH ^ ! Ω PC = dp µ

• Extended polymomentum phase space : ( y a , p ⌫ a , x ⌫ ) =: ( z v , x µ ) = z M Z : V n 1 ( Y ) / V n 0 ( Y ) ! Y. • Canonical structure on Z : mod V n a dy a ^ ! µ Θ := [ p µ 0 ( Y )] • Polysymplectic form mod V n +1 Ω := [ d Θ ( Y )] 1 Ω = � dy a ^ dp µ a ^ ! µ • DW equations in geometric formulation: n Ω = dH X 11

Hamiltonian multivector fields and Hamiltonian forms X 2 V p TZ , is called vertical if p • A multivector field of degree p , X F = 0 for any form F 2 V ⇤ p 0 ( Z ) . • The polysymplectic form establishes a map of horizontal p � forms F 2 V p p n � p 0 ( Z ) to vertical multivector fields of degree ( n � p ) , X F , called Hamiltonian : n � p p X F Ω = d F. • The forms for which the map (2) exists are called Hamiltonian . • The natural product operation of Hamiltonian forms is the co- exterior product ^ p + q � n p q p q F := ⇤ � 1 ( ⇤ F • F ^ ⇤ F ) 2 ( Z ) 0 • co-exterior product is graded commutative and associative.

Poisson-Gerstenhaber brackets • P-G brackets: p q } = ( � 1) ( n � p ) n � p F 2 = ( � 1) ( n � p ) n � p q n � q { [ F 1 , F 2 ] X 1 d X 1 X 2 Ω ^ p + q � n +1 2 ( Z ) . 0 • The space of Hamiltonian forms with the operations { [ , ] } and • is a (Poisson-)Gerstenhaber algebra, viz. p q q p } = � ( � 1) g 1 g 2 { { [ F, F ] [ F, F ] } , p q r q r p ( � 1) g 1 g 3 { } + ( � 1) g 1 g 2 { [ F, { [ F, F ] } ] [ F, { [ F, F ] } ] } p q r ( � 1) g 2 g 3 { + [ F, { [ F, F ] } ] } = 0 , p q p q F + ( � 1) g 1 ( g 2 +1) q p r r r { [ F, F • F ] } = { [ F, F ] } • F • { [ F, F ] } , g 1 = n � p � 1 , g 2 = n � q � 1 , g 3 = n � r � 1 .

Applications of P-G brackets • The pairs of ”canonically conjugate variables”: [ p µ a ! µ , y b ] } = � b [ p µ a ! µ , y b ! ⌫ ] } = � b [ p µ a , y b ! ⌫ ] } = � b a � µ { a , { a ! ⌫ , { ⌫ . • DW Hamiltonian equation in the bracket form: d • F = � � ( � 1) n { } + d h • F, [ H, F ] for Hamiltonian ( n � 1) � form F := F µ ! µ ; p 1 ( n � p )! @ M F µ 1 ... µ n � p @ µ z M dx µ • @ µ 1 ... µ n � p d • F := ! , p 1 ( n � p )! @ µ F µ 1 ... µ n � p dx µ • @ µ 1 ... µ n � p d h • F := ! , � = ± 1 for the Euclidean/Minkowskian signature of X . p } + d h F. • More general: d F = { [ H ! , F ]

Application to quantization of fields • Geometric prequantization of P-G brackets. Prequantization map F ! O F acting on (prequantum) Hilbert space fulfills three prorties: (Q1) the map F ! O F is linear; (Q2) if F is constant, then O F is the corresponding multi- plication operator; (Q3) the Poisson bracket of dynamical variables is related to the commutator of the corresponding operators: [ O F 1 , O F 2 ] = � i ~ O { F 1 ,F 2 } , [ A, B ] := A � B � ( � 1) deg A deg B B � A.

• Explicit construction of prequantum operator of form F : O F = i ~ [ X F , d ] + ( X F Θ ) • + F • is a inhomogeneous operator, acts on prequantum wave func- tions Ψ ( y, p, x ) – inhomogeneous forms on the polymomentum phase space. • “Prequantum Schr¨ odinger equation” X 0 Ω MS = 0 ! O 0 Ψ = 0 ) i � ~ d • Ψ = O H ( Ψ ) • Polarization: Ψ ( y, p, x ) ! Ψ ( y, x ) . • Normalization of prequantum wave functions leads to the metric structure on the space-time! ) Co-exterior algebra ! Clifford algebra.