Non-uniqueness of quantization, reality conditions, complex time - PowerPoint PPT Presentation

Non-uniqueness of quantization, reality conditions, complex time evolution and coherent state transforms Jos´ e Mour˜ ao Mathematics Department, T´ ecnico Lisboa Univ Lisboa 9th Mathematical Physics Meeting Belgrade M ∩ Φ September 18 – 23, 2017, On work in collaboration with Jo˜ ao P. Nunes

Summary 1. Ambiguity of quantization and preferred observables.   F = ( F 1 , . . . , F n ) � ( ω, J F , γ F )   � �  H Q Ψ = ψ ( F ) e − k F , || Ψ || < ∞ ⊂ H prQ = ( M, ω ) � F   F prQ | H Q �  = F F �→  F 2. Geometry on the (infinite dimensional) space of K¨ ahler structures ( ⊂ space of quantizations), complex time evolution and Coherent State Transforms. 2

1. Ambiguity of quantization and preferred observables 1.1 Introduction With > 100 years of General Relativity and > 90 years of Quantum Mechanics it is becoming increasingly embarassing the fact that there is not a fully consistent theory of Quantum Gravity. The strongest candidates to succeed, String Theory and Loop Quantum Gravity (LQG), continue facing conceptual and technical problems. One of the problems one is faced with and the one we will address today is that of nonuniqueness of quantization of a classical system. 3

The dream of the founders of quantum mechanics was to have quantization as a well defined process assigning a quantum sys- tem to every classical system and satisfying the correspondence principle Quantization Functor (?) : ( M, ω ) �→ Q � ( M, ω ) � → 0 �→ ( M, ω ) It was soon realized that this can never be the case even for the simplest systems. 4

Particle in the line ( 1 dof) Classical ( R 2 , dp ∧ dq ) , ( M, ω ) = ∂f ∂q − ∂f ∂ ∂ f � X f = ∂p ∂q ∂p Quantum Q Sch ( R 2 , dp ∧ dq ) : � H Q L 2 ( R , dq ) = Sch Q Sch ( q ) = � q �→ q = q � p = i � ∂ Q Sch ( p ) = � p �→ � ∂q f ( q, p ) �→ ?? H = − � 2 H = 1 ∂ 2 p 2 + V ( q ) ( H ) = � Q Sch �→ ∂q 2 + V ( q ) � 2 H Q H Q = q Sch 5

Groenewold (1946) – van Hove (1951) no go Thm: It is impossible, even for systems with one degree of freedom, to quantize all observables exactly as Dirac hoped � Q � ( f ) = f i � Q � ( { f, g } ) [ Q � ( f ) , Q � ( h )] = and satisfy natural additional requirements like irreducibility of the quantiza- tion. In order to quantize one needs to add additional data to the classical system. eg choose a (sufficiently big but not too big ...) (Lie) subalgebra of the algebra of all observables A = Span C { 1 , q, p } Then we have to study the dependence of the quantum theory on the addi- tional data. 6

1.2 Geometric Quantization Geometric quantization is mathematically perhaps the best defined quantiza- tion 1 2 π � [ ω ] ∈ H 2 ( M, Z ) ( M, ω ) , Prequantum data: ( L, ∇ , h ), L → M, F ∇ = ω � Pre-quantum Hilbert space: � � � h ( s, s ) ω n H prQ = Γ L 2 ( M, L ) = s ∈ Γ ∞ ( M, L ) : || s || 2 = n ! < ∞ M f prQ = i � ∇ X f + f Pre-quantum observables: � ( f ) = � f = Q prQ � This almost works! But the Hilbert space is too large, the representation is reducible. We need a smaller Hilbert space: Prequantization ⇒ Quantization 7

Additional Data in Geometric Quantization Generalizing what is done in the Schr¨ odinger representation, for systems with one degree of freedom, to fix a quantization one chooses (locally) a preferred observable – F ( q, p ) ∗ – and then works with wave functions of the form � � Ψ ∈ H prQ : ∇ X F Ψ = 0 , || Ψ || < ∞ H prQ � H Q = = F � � Ψ( q, p ) = ψ ( F ) e i G ( q,p ) , || Ψ || < ∞ ⊂ H prQ = on which the preferred observable acts diagonally Q F F prQ | H Q � ( F ) = � = F. F ∗ for systems with n degrees of freedom one chooses (locally) n independent observables in involution F 1 , . . . , F n , { F j , F k } = 0. The distribution P = < X F j , j = 1 , . . . n > is called polarization associated with this choice. 8

(Non–)Equivalence of different Quantizations Are all these quantizations (for different choices of F ) physically equivalent? NO! Consider the observable: H λ = p 2 2 + q 2 2 + λ q 4 4 , λ ≥ 0 and let Sp Sch ( H λ ) denote the (discrete) spectrum of H λ in the Schr¨ odinger quantization, i.e. the spectrum of the operator ( H λ ) = − � 2 ∂q 2 + q 2 ∂ 2 2 + λq 4 Q Sch � 2 4 acting on H Q Sch = L 2 ( R , dq ). 9

Now consider the 1–parameter family of quantizations with Hilbert spaces H Q H λ for which the role of preferred observable is played by H λ . Then, one finds that � Ψ( q, p ) : ∇ X Hλ Ψ = 0 � H Q = = H λ � Ψ( q, p ) = ψ ( H λ ) e i G λ ( q,p ) � = = � ∞ � � ψ n δ ( H λ − E λ n ) e iG λ ( q,p ) = , (1) n =0 where E λ n are defined by the Bohr-Sommerfeld conditions � pdq = � ( n + 1 2) . (2) H λ = E λ n Since H λ acts diagonally on this quantization we conclude from (1) that its spectrum in this quantization is given by (2) Sp H λ ( H λ ) = { E λ n , n ∈ N 0 } . It is known that on one hand Sp Sch ( H 0 ) = Sp H 0 ( H 0 ) but on the other hand Sp Sch ( H λ ) � = Sp H λ ( H λ ) for all λ > 0 so that the two quantizations Q Sch and � X Hλ are physically inequivalent if λ > 0! Wins Q Sch Q ! � � 10

1.3 Ambiguity of quantization and reality conditions LQG is facing a similar problem with the Ashtekar–Barbero con- nection as preferred observable A β = Γ( E ) + β K ⇒ Ψ β ( E, K ) = ψ ( A β ) e iG β ( E,K ) . Are the quantizations based on the choice of connections with different (Immirzi) parameters equivalent? No, because they lead to different spectra of the area operator. Here it is less obvious which one is the ”correct”one. Studies of the black hole entropy formula seemed to indicate the value √ β = ln(3) / 8 π ?? 11

Other, recent studies (e.g. Pranzetti, Sahlmann, Phys Lett 2015, Ben Achour, Livine, arXiv:1705.03772) however seem to point back to β = √− 1. This corresponds to the Ashtekar con- nection √ A √− 1 = Γ + − 1 K The study of quantizations based on compex valued observables like this has been the focus of most of our recent work. It turns out that for some choices of complex observables quanti- zation is in fact mathematically better defined then quantization based on real observables and this may help addressing some of the technical issues faced by LQG. 12

Complex observables and reality conditions: rescued by the power of complex analysis Let us illustrate the general situation with a one degree of free- dom system. Consider the quantum observable dz f ∧ dz f = − 2 if ′ ( p ) dq ∧ dp . z f = q + if ( p ) , It turns out that if f ′ ( p ) > 0 then several remarkable simplifying facts occur: 13

F f = z f = q + if ( p ) 1. Complex Structure: There is a unique complex structure J f on R 2 for which z f is a global holomorphic coordinate. 2. K¨ ahler Metric: The symplectic form together with the com- plex structure J f define on R 2 a K¨ ahler metric 1 f ′ ( p ) dq 2 + f ′ ( p ) dp 2 γ f = � � ′′ 1 R ( γ f ) = − . f ′ ( p ) 14

3. Quantum Hilbert space much better defined than in the case of quantizations based on real observables: � � H Q Ψ( q, p ) = ψ ( z f ) e − k f ( p ) , || Ψ || < ∞ X zf = where ψ is a J f –holomorphic function and � f ( p ) dp is a K¨ k f ( p ) = pf ( p ) − ahler potential. 4. The inner product is not ambiguous and it fixes the reality conditions: � R 2 ψ 1 ( z f ) ψ 2 ( z f ) e − 2 k f ( p ) dqdp . < Ψ 1 , Ψ 2 > = 15

2. Generalized Coherent State Transforms 2.1 Imaginary time: why?? It is precisely to study the dependence of Q � on the choice of preferred complex observables that evolution in imaginary time enters the scene. � ∂q : q �→ q + t f ( p ) t � √− 1 s √ f ( p ) dp � X H = f ( p ) ∂ H = �→ q + − 1 s f ( p ) Imaginary time evolution is not new in quantum mechanics. Many amplitudes can be obtained by making the famous (but misterious) Wick rotation: t � is – e.g. semiclassical probabilites of tunneling given by imaginary time evolution. What we are studying is a new way of looking at imaginary (or complex) time evolution in (some situations in) quantum mechanics and in geometry. 16

In K¨ ahler geometry imaginary time evolution leads to geodesics in the (infinite dimensional) space of K¨ ahler metrics ( ⊂ quanti- zations) in a given cohomology class, and is used to study the stability of polarized varieties [Semmes, Donaldson, Tian]. In loop quantum gravity complex time Hamiltonian evolution was proposed by Thiemann in ’96 in order to transform the spin connection into the Ashtekar connection. Γ �→ A i = Γ + iK. 17

2.2 Generalized Coherent State Transforms (CST) So we can use one parameter groups of complex canonical trans- formations to move in the space of quantizations T , parametrized by choices of preferred observables (e.g. K¨ ahler structures), e τ L XH : P 0 = < X F 1 , . . . , X F n > P τ = e τ L XH P 0 = (3) �→ = < X e τXH ( F 1 ) , . . . , X e τXH ( F n ) > In the present section we will see how to lift this action to the “quantum bundle”over the space of quantizations, H Q − → T , in order to relate different quantizations : H Q → H Q V H P 0 − (4) τ P τ 18