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1. The Origin of Complex Structures in QFT on Curved Space-Times Philip Tillman Tuesday, January 8, 2008 P. Tillman () The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 1 / 16

2. Outline 1 Technical Di¢ culties 2 Complex Structures and Time Evolution 3 Construction of the Fock Space 4 The Unruh E¤ect 5 The Quantum Field and Particles 6 Comments/Open Questions P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16

8. Technical Di¢ culties (1) 1 No Poincare Symmetry INo "preferred" time and No global Fourier transforms 2 Unboundedness of operators: [ˆx, ˆp] = i h m ei ˆx ei ˆp = e [ˆx,ˆp]/2 eˆx+ˆp | {z } Weyl Relations + strong continuity 3 Operator-valued distributions ˆφ (x) are singular ISmear them out by test functions f in ˆφ (f ) P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 3 / 16

11. Technical Di¢ culties (2) 4. Well-posed initial value formulation ISolutions of the …eld equations are well-de…ned everywhere in space-time IDe…ne space-time to be globally hyperbolic, M = Σ R 5. Unitary Inequivalent constructions of Hilbert spaces 6. Renormalization 7. Relearn or entirely unlearn basic notions of QFT IE.g.) The "particle notion" must be unlearned P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 4 / 16

15. Complex Structures and Time Evolution The Real Scalar (KG) Field: rµrµ + m2 φ = 0 The Schrödinger equation for a single particle (φ+) in a curved space-time is the de…nition of ˆH: ˆHφ+ def = iLX φ+ LX is the Lie derivative in the Xa (time) direction, h = 1 The complex structure (Ashtekar-Magnon): J def = LX / ( LX LX )1/2 , J2 = 1 P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 5 / 16

16. Complex Structures and Time Evolution Splitting a solution φ into positive/negative frequency solutions: φ = φ+ + φ Jφ = iφ Positive frequencies are particles states Negative frequencies are anti-particles states E.g.) Minkowski Space LX = ∂/∂τ Jφ = J u (x) e iωτ + u (x) e+iωτ = (+i) u (x) e iωτ + ( i) u (x) eiωτ The decomposition φ = φ+ + φ and is related to the choice of J. P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 6 / 16

17. Construction of the Fock Space Given a single particle Hilbert space H then de…ne the Fock space F: F def = ?|{z} vacuum H|{z} single particle H H| {z } two particle The construction of H is not complete. . . We need an inner-product h, iH for expectation values. F The choice of H is also related to choice of J because H is the space of all φ+ and Jφ+ = +iφ+. P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 7 / 16

18. The Symplectic Structure To construct an inner product we use the symplectic structure Ω ( , ). Let S be the set of all fΦ = (φ, π)g that satisfy the …eld equations (the space of solutions). Fact due the form of the KG …eld equation: S is a symplectic vector space. Ω (Φ1, Φ2) = Z Σ0 (π1φ2 π2φ1) d3 x = Z Σ0 ((raφ1) φ2 (raφ2) φ1) na p hd3 x h is the spacial metric on the Cauchy surface Σ0. P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 8 / 16

19. The Inner Product The single particle Hilbert space H inner product: 2 hΦ1, Φ2iH def = Ω (JΦ1, Φ2) iΩ (Φ1, Φ2) H is de…ned by the Cauchy completion Sµ of S in the positive de…nite norm: µ (Φ1, Φ2) def = Ω (JΦ1, Φ2) We now can de…ne the Fock space F: F def = ?|{z} vacuum H|{z} single particle H H| {z } two particle P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 9 / 16

20. What We Have Learned So Far 1 The choice of time evolution (Lie derivative in X-direction) is directly related to the choice of complex structure: J def = LX / ( LX LX )1/2 , J2 = 1 2 Also the choice of J de…nes our single particle Hilbert space (hence our Fock space): H def = φ+j φ+ 2 Sµ, Jφ+ = +iφ+ , h, iH : H ! C where Sµ is the closure of S in µ ( , ) = Ω (J , ). 3 Next, an interesting physical consequence. . . P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 10 / 16

23. The Unruh E¤ect In Minkowski space take 2 observers with particle detectors. An inertial observer, and a uniformly accelerating (Rindler) observer: PICTURE The inertial observer measures the state of the universe on Σ to be the (Minkowski) vacuum j?Mi at point p. P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 11 / 16

24. The Unruh E¤ect The Rindler observer de…nes positive frequency particles di¤erently: φ = φ (R) + + φ (R) as opposed to: φ = φ (M) + + φ (M) Two quantizations Q1 and Q2: φ Q1 ! ˆφ1 = ˆφ (M) 1+ + ˆφ (M) 1 Q2 & ˆφ2 = ˆφ (R) 2+ + ˆφ (R) 2 P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 12 / 16

25. The Unruh E¤ect Bogoliubov transformation between the two: U ˆφ1U 1 = ˆφ2 Consequently in the state j?Mi, the Rindler observer measures Rindler particles φ (R) + : U j?Mi = c j?R i j2 particlesi j4 particlesi Conundrum: The Rindler observer will measure particles (ˆφ (R) 2+ ) when the inertial observer would measure none (vacuum j?Mi) Explanation: Each observation of a Rindler particle, is "seen" as an emission of a Minkowski particle by the inertial observer (Unruh and Wald). Weird! P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 13 / 16

28. The Unruh E¤ect Why isn’t U j?Mi = j?R i? Because positive frequency modes mix with negative ones Bk 6= 0: φ (M) + = ∑ k Ak φ (R) +k + Bk φ (R) k Why aren’t all de…nitions of vacuums the equivalent? Ashtekar-Magnon: Ask the opposite question. Why is the de…nition the same for all inertial observers? A: Because of a time-like Killing symmetry ) positive and negative modes don’t mix P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 14 / 16

33. The Quantum Field and Particles The Relativity Principle: All observers are equivalent, including non-inertial and inertial. Consequences: No "universal" de…nition of particles and No "universal" de…nition of vacuum Need observer dependent de…nition of particles and vacuums state. The …eld is more fundamental than particles: The "particle notion" exists in limited circumstances. Haag: The "quantum …eld" concept is the encorporation of the principle of locality into quantum theory. P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 15 / 16

38. Comments/Open Questions 1 This type of construction only works for free …elds. 1 Free …elds are linear …elds, for nonlinear …elds no Ω exists on S. 2 Examples of nonlinear …elds: interacting …elds, gauge …elds, and gravity 3 Renormalization in curved space-time has been formulated by Fredenhagen-Brunetti and Hollands-Wald. 2 It does not consider back reaction: Gµν = 8π ˆTµν I.e. the e¤ect on geometry by these …elds. IWe treat the geometry as …xed when it isn’t really. 3 The BIG question: quantum gravity? P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16

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