HamiltonJacobi meets M obius Alon E. Faraggi AEF, Marco Matone, - PowerPoint PPT Presentation

Hamilton–Jacobi meets M¨ obius Alon E. Faraggi • AEF, Marco Matone, PLB 450 (1999) 34; ... ; IJMPA 15 (2000) 1869. • G. Bertoldi, AEF & M. Matone, CQG 17 (2000) 3925. • AEF, Marco Matone, EPJC 74 (2014) 2694. • AEF, AHEP. 2013 (2013) 957394 ; 1305.0044. related: Edward Floyd 1982–2011; Robert Wyatt; Bill Poirier. DISCRETE2014, King’s College, London, 5 December 2014

• Motivation – quantum gravity • The Equivalence Postulate ⇒ QSHJE → Schr¨ odinger eq. • The Equivalence Postulate ⇒ CoCyCle Condition → M¨ obius invariance • Phase space duality & Legendre transformations • The Equivalence Postulate ⇒ Energy quantization & Time Parameterisation • M¨ obius invariance ⇒ Compact universe • Conclusions

Motivation General Relativity: Covariance & Equivalence Principle → fundamental geometrical principle Quantum Mechanics: No Such Principle Axiomatic formulation ... P ∼ | Ψ | 2 However Quantum + Gravity Theory not known Main effort: quantize GR; quantize space–time: e.g. superstring theory The main successes of string theory: 1) Viable perturbative approach to quantum gravity 2) Unification of gravity, gauge & matter structures i.e. construction of phenomenologically realistic models → relevant for experimental observation State of the art: MSSM from string theory (AEF, Nanopoulos, Yuan, NPB 335 (1990) 347) (Cleaver, AEF, Nanopoulos, PLB 455 (1999) 135)

Other approaches Geometrical Greene, Kirklin, Miron, Ross (1987) Donagi, Ovrut, Pantev, Waldram (1999) Blumenhagen, Moster, Reinbacher, Weigand (2006) Heckman, Vafa (2008) ........ Orbifolds Ibanez, Nilles, Quevedo (1987) Bailin, Love, Thomas (1987) Kobayashi, Raby, Zhang (2004) Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange, Wingerter (2007) Blaszczyk, Groot–Nibbelink, Ruehle, Trapletti, Vaudrevange (2010) ....... Other CFTs Gepner (1987) Schellekens, Yankielowicz (1989) Gato–Rivera, Schellekens (2009) ....... Orientifolds Cvetic, Shiu, Uranga (2001) Ibanez, Marchesano, Rabadan (2001) Kiristis, Schellekens, Tsulaia (2008) .......

Adaptation of Hamilton–Jacobi theory q = ∂H p = − ∂H Hamilton’s equations of motion ˙ , ˙ ∂p ∂q Q = ∂K P = − ∂K ˙ ∂P ≡ 0 , ˙ H ( q, p ) − → K ( Q, P ) ≡ 0 = ⇒ ∂Q ≡ 0 The solution is the Classical Hamilton–Jacobi Equation → K ( Q, P ) = H ( q, p = ∂S ∂q ) + ∂S H ( q, p ) − ∂t = 0 ⇒ CHJE � 2 � ∂S 0 1 stationary case − → + V ( q ) − E = 0 2 m ∂q ( q, p ) → ( Q, P ) via canonical transformations q, p are independent. Solve. Then p = ∂S ∂q

Quantum mechanics: [ˆ q, ˆ p ] = i � → q, p → not independent Assume H → K i.e. W ( Q ) = V ( Q ) − E = 0 always exists p = ∂S But q, p not independent. ∂q . Equivalence postulate: Consider the transformations on p = ∂ ˜ ( q , S 0 ( q ) , p = ∂S 0 S 0 q , ˜ ∂q ) − → ( ˜ S 0 (˜ q ) , ˜ q ) ∂ ˜ Such that ˜ W ( q ) − → W (˜ q ) = 0 exist for all W ( q ) = ⇒ QHJE − → Schr¨ odinger equation

Implies: Covariance of HJE � 2 � ∂S 0 1 But: + V ( q ) − E = 0 2 m ∂q Is not covariant under q → q ( q ) . ˜ Further: W ( q ) ≡ 0 is a fixed state under q → q ( q ) . ˜ � 2 � ∂S 0 1 Assume: + W ( q ) + Q ( q ) = 0 2 m ∂q � − 2 � ∂ ˜ q ˜ The most general transformations W (˜ q ) = W ( q ) + (˜ q ; q ) , ∂q � − 2 � ∂ ˜ q ˜ Q (˜ q ) = Q ( q ) − (˜ q ; q ) , ∂q ˜ with S 0 (˜ q ) = S 0 ( q ) under q → ˜ q = ˜ q ( q )

W 0 ( q 0 ) = 0 W ( q ) = ( q ; q 0 ) With: → All: b b W (q ) a a c c W (q ) W (q ) � 2 � � ∂q b � ( q a ; q c ) = ( q a ; q b ) − ( q c ; q b ) Cocycle Condition: ∂q c ⇒ Theorem ( q a ; q c ) invariant under M¨ obius transformations γ ( q a ) In 1D: ( q a ; q c ) { q a ; q c } ∼ Uniquely � 2 � 2 � ∂y � ∂y Schwarzian derivative { h ( x ); x ( y ) } = { h ( x ); y } − { x ; y } . ∂x ∂x U ( q ) = { h ( q ); q } = { Ah + B Ch + D ; q } Invariant under M¨ obius transformations

� 2 = β 2 i 2 S 0 � ∂S 0 � � β ; q } − { S 0 ; q } Identity { e ∂q 2 Make the following identifications W ( q ) = − β 2 i 2 S 0 β ; q } = V ( q ) − E 4 m { e Q ( q ) = β 2 4 m { S 0 ; q } The Modified Hamilton–Jacobi Equation becomes � 2 + V ( q ) − E + β 2 1 � ∂S 0 4 m { S 0 ; q } = 0 2 m ∂q ± β ˜ QM : W (˜ q ) ≡ V (˜ q ) − E ≡ 0 ⇒ S 0 = 2 ln ˜ q � = A ˜ q + B

From the properties of the SD { ; } V ( q ) − E = − β 2 i 2 S 0 β ; q } 4 m { e is a potential of the 2 nd –order diff. Eq. � � − β 2 ∂ 2 ∂q 2 + V ( q ) − E Ψ( q ) = 0 ⇒ β = � 2 m 1 A e + i � S 0 + B e − i � � � S 0 The general solution Ψ( q ) = � S ′ 0 = e iα w + i ¯ e + i 2 S 0 ℓ w = ψ 1 and � w − iℓ ψ 2 ℓ = ℓ 1 + i ℓ 2 ℓ 1 � = 0 α ∈ R

The equivalence transformation ˜ W ( q ) = V ( q ) − E − → W (˜ q ) = 0 always exists q = ψ 1 We have to find q → ˜ q take ˜ ψ 2 � � − β 2 ∂ 2 ∂ 2 q 2 ˜ then ∂q 2 + V ( q ) − E Ψ( q ) = 0 → ψ (˜ q ) = 0 2 m ∂ ˜ � − 1 � dq 2 ˜ where ψ (˜ q ) = ψ ( q ) d ˜ q

Generalizations: 0 ( q v ) , p v = ∂S v ( q , S 0 ( q ) , p = ∂S 0 ( q v , S v 0 Under ∂q ) − → ∂q v ) , p v = ∂S v ( q v ) = ∂S ( q ) ∂S ( q ) ∂q i ij = ∂q i , = J v p, where J v � = ∂q v ∂q v ∂q v ∂q v ∂q i j j j j i | p | 2 = p vT p v p T p = p T J vT J v p ( p v | p ) ≡ | p v | 2 with . p T p � p c | p b � � � Cocycle condition → ( q a ; q c ) = ( q a ; q b ) − ( q c ; q b ) . invariant under D–dimensional Mobi¨ us (conformal) trans.

Quadratic identity: α 2 ( ∇ S 0 ) · ( ∇ S 0 ) = ∆( Re αS 0 ) � � − ∆ R 2 ∇ R · ∇ S 0 R − α + ∆ S 0 , Re αS 0 R or α 2 ( ∂S ) · ( ∂S ) = ∂ 2 ( Re αS ) − ∂ 2 R � 2 ∂R · ∂S � + ∂ 2 S R − α , Re αS R or α 2 ( ∂S − eA ) · ( ∂S − eA ) = D 2 ( Re αS ) − ∂ 2 R R − α � � R 2 ( ∂S − eA ) R 2 ∂ · , Re αS D µ = ∂ µ − αeA µ

Phase space duality & Legendre transformations intimate connection between p − q duality & the equivalence postulate q = ∂H − ∂H Hamilton’s Eqs. ˙ , p = ˙ ∂p ∂q invariant under p − → − q 2 m p 2 + V ( q ) − E = 0 1 breaks down once V ( q ) is specified e.g. Aim Formulation with manifest p − q duality p = ∂S q = ∂T recall define ∂q ∂p S = p∂T T = q∂S ∂p − T , ∂q − S S 0 = p∂T 0 T 0 = q∂S 0 Stationary Case: ∂p − T 0 , ∂q − S 0

Invariant under M¨ obius transformations: → q v = Aq + B q − Cq + D, = ρ − 1 ( Cq + D ) 2 p , p − → p v ρ = AD − BC 0 ( p v ) = T 0 ( p ) + ρ − 1 ( ACq 2 + 2 BCq + BD ) p. → T v T 0 − q → q v = v ( q ) S v 0 ( q v ) = S 0 ( q ) Transformations: defined by ( S 0 scalar function under v ) Associate a 2 nd order diff. eq. with the Legendre transformation: � � q √ p � ∂ 2 � + U ( S 0 ) = 0 √ p ∂S 2 0 � 2 q ′′′ � q ′′ U ( S 0 ) = 1 q ′ − 3 where 2 { q, S 0 } q ′ 2

We can derive this eq. in several ways p = ∂S 0 p ∂q ⇒ = 1 ∂S 0 ∂q ∂ ∂S 0 + p ∂ 2 q ∂p ∂q ⇒ = 0 ∂S 2 ∂S 0 ∂S 0 0 S 0 q √ p √ p ∂ 2 ∂ 2 S 0 rewritten as q √ p = √ p = − U ( S 0 ) ∂ 2 S 0 ( q ) = 1 √ p ∂T 0 or : ∂ √ p − T 0 ∂S 2 2 0 � � q √ p � ∂ 2 � = ⇒ + U ( S 0 ) = 0 √ p ∂S 2 0

manifest p ↔ q – S 0 ↔ T 0 duality with p = ∂S 0 q = ∂T 0 ∂q ∂p S 0 = p∂T 0 T 0 = q∂S 0 ∂p − T 0 ∂q − S 0 � � q √ p � � p √ q � � ∂ 2 ∂ 2 � � + U ( S 0 ) = 0 + V ( T 0 ) = 0 √ p √ q ∂S 2 ∂T 2 0 0 Involutive Legendre transformation ↔ duality

Self–dual states States with pq = γ = const are simultaneous solutions of the two pictures with S 0 = − T 0 + const S 0 ( q ) = γ ln γ q q T 0 ( p ) = γ ln γ p p S 0 + T 0 = pq = γ where γ q γ p γ = e pq = γ self–dual states γ sd = ± � W sd = W 0 = 0 self–dual states 2 i

Energy quantization: ⇒ (Ψ , Ψ ′ ) continuous ; Ψ ∈ L 2 ( R ) Probability: = = ⇒ quantization, bound states What are the conditions on the trivializing transformations? = ψ D q 0 = w = ψ 1 ψ 2 ψ − 4 m we have { w, q } = � 2 ( V ( q ) − E ) w � = const ; w ∈ C 2 ( R ) and w ′′ differentiable on R ⇒ { w, q − 1 } = q 4 { w, q } In addition from the properties of { , } → R ) and w ′′ differentiable on ˆ w � = const ; w ∈ C 2 ( ˆ ⇒ R ˆ where R = R ∪ {∞}

= ⇒ ⇒ continuity of ( ψ D , ψ ) and ( ψ D ′ , ψ ′ ) Equivalence postulate = Theorem:  P 2 − > 0 for q < q −  if V ( q ) − E = P 2 + > 0 for q > q +  then the ratio w = ψ D /ψ is continuous on ˆ R odinger equation admits an L 2 ( R ) solution iff the Schr¨

 0 | q | ≤ L  Potential Well: V ( q ) = V 0 | q | > L  √ √ 2 m ( V 0 − E ) 2 mE k = K = � � Ψ 1 Ψ 1 | q | ≤ L 1 = cos kq 2 = sin kq Ψ 2 1 = e − Kq Ψ 2 2 = e Kq q > L Ψ , Ψ ′ continuous take (1 , 1) : ⇒ k tan kL = K  cos(2 kL ) − e − 2 K ( q + L ) q < − L    1  ⇒ w = sin(2 kL ) tan( kq ) | q | ≤ L [ k sin(2 kL )]  e 2 K ( q − L ) − cos(2 kL )  q > L  