Osterwalder-Schrader Stability

Transcription

1 OS Stability p.1/57 Osterwalder-Schrader Stability Wayne Polyzou - Victor Wessels polyzou@uiowa.edu The University of Iowa

2 OS Stability p.2/57 Outline Time-Ordered Green Functions The Bethe-Salpeter Equation The Mandelstam-Method - Physics Quasi-Schwinger Functions - (analytic continuation) Quasi-Wightman Functions - (analytic continuation)

3 OS Stability p.3/57 Outline OS Reconstruction of RQM Reflection Positivity Stability and the BS equation Null Spaces The Instability

4 OS Stability p.4/57 Time-ordered Green Functions G(x 1,, x n ) := 0 T (φ(x 1 ) φ(x n )) 0 T (φ(x 1 )φ(x 2 )) := θ(t 0 1 t 0 2)φ(x 1 )φ(x 2 ) ± θ(t 0 2 t 0 1)φ(x 2 )φ(x 1 ).

5 OS Stability p.5/57 Cluster Decomposition I = Π p W 2 (x 1, x 2 ) := 0 φ(x 1 )φ(x 2 ) 0 = 0 φ(x 1 )( Π p )φ(x 2 ) 0 = 0 φ(x 1 ) 0 0 φ(x 2 ) φ(x 1 )Π p φ(x 2 ) 0 W 2 (x 1, x 2 ) = W 1 (x 1 )W 1 (x 2 ) + W t (x 1, x 2 )

6 OS Stability p.6/57 Cluster Decomposition W 2 (x 1, x 2 ) = W 1 (x 1 )W 1 (x 2 ) + W t (x 1, x 2 ) W t (x 1, x 2 ) d 3 p 2ω(p) 0 φ(0) p eip (x 1 x 2 ) p φ(0) 0 lim W t(x 1 x 2 ) 0 (x 1 x 2 ) 2

7 OS Stability p.7/57 Cluster Decomposition for G 4 G 1 (x) = constant = 0 G 4 (x 1, x 2, x 3, x 4 ) = G 2 (x 1, x 4 )G 2 (x 2, x 3 ) + G 2 (x 1, x 3 )G 2 (x 2, x 4 )+ G 2 (x 1, x 2 )G 2 (x 3, x 4 ) + G t (x 1, x 2, x 3, x 4 ) G 4 = G 0 + G t

8 OS Stability p.8/57 Bethe-Salpeter Equation Treat G 4 as the kernel of an integral operator K := G 1 0 G 1 4 G 4 G 0 = G 0 KG 4 = G t BS equation determines G t from G 0 and K

9 OS Stability p.9/57 Homogeneous BS equation G 4 (x 1, x 2, x 2, x 1) G 4(X X, x, x ) : X = 1 2 (x 1 + x 2 ) x = x 1 x 2 X = 1 2 (x 1 + x 2) x = x 2 x 1

10 OS Stability p.10/57 Homogeneous BS equation G 4(X X, x, x ) = θ(x 0 X 0 ) 0 T (φ(x + x 2 )φ(x x 2 )) T (φ (X + x 2 )φ (X x )) 0 +other time orderings 2

11 OS Stability p.11/57 Homogeneous BS equation 0 T (φ(x+ x 2 )φ(x x 2 ))T (φ (X + x 2 )φ (X x 2 )) 0 = 0 T (φ( x 2 )φ( x 2 )) p n e ip n (X X ) d 3 p n 2ω n n (p) p n T (φ ( x 2 )φ ( x 2 )) 0

12 OS Stability p.12/57 Homogeneous BS equation Use existence of complete sets of physical states and the representation θ(x 0 ) = 1 2πi ds eisx0 s i0 +

13 OS Stability p.13/57 Homogeneous BS equation 1 2πi d 4 p χ P n (x)χ P n (x ) P 0 ω n i0 +e ip (X X ) + χ Pn (x) := 0 T (φ( 1 2 x)φ( 1 2x)) p, n 2ωn ( p ) χ P n (x ) := p, n T (φ ( 1 2 x )φ ( 1 2 x )) 0 2ωn ( p ).

14 OS Stability p.14/57 Homogeneous BS equation G P (x, x ) = 1 (2π) 2 d 4 Xe ip X G (X, x, y) = 2πi n χ Pn (x)χ P n (x ) P 0 ω n i0 + +

15 OS Stability p.15/57 Homogeneous BS equation lim (P 0 ω n ) P 0 ω n G 1 P (x, y) G P (y, x ) = 0 d 4 x G 1 P n (x, x )χ Pn (x ) = 0

16 OS Stability p.16/57 BS normalization condition P 0 ( (P 0 ω n ) ) G 1 P (x, y) G P (y, x )d 4 y G 1 P 1 = 2πi(χ Pn P χ 0 P n )

17 OS Stability p.17/57 Mandelstam and Physics Goal: compute p n O(y) p m Input: G 4 (x 1, x 2, x 3, x 4 ) := 0 T ( φ(x 1 ) φ (x 4 ) ) 0 G 5 (x 1, x 2, y, x 3, x 4 ) := 0 T ( φ(x 1 )φ(x 2 )O(y)φ (x 3 )φ (x 4 ) ) 0

18 OS Stability p.18/57 Mandelstam Using the same methods with G 5 gives ( 2πi) 2 G 5 (p, x, 0, q, x ) = χ Pn (x) p n O(0) p m χ P m (x ) (P 0 ω n i0 + )(P 0 ω m i0 + ) +

19 OS Stability p.19/57 Mandelstam The desired physical matrix element can be extracted using the normalization condition p n O(y) p m = lim(p 0 ω n )(P 0 ω m) (χ P n, G 1 P P 0 G 5 (p,, y, p, ) G 1 P P 0 χ P m )

20 OS Stability p.20/57 Mandelstam For good Green functions this method can be generalized to extract any matrix element of O. Assumes that model behaves as if it has physical intermediate states. Consistency between G 4 and G 5 required. The underlying quantum theory has disappeared.

21 OS Stability p.21/57 Quasi-Schwinger Functions Analyticity of G n in times follows from spectral properties: ˆf(t) = 0 e iet f(e)de supp(f) [0, ) ˆf(t) ˆf(t + iτ) τ > 0 ˆf(t) ˆf(e iφ t) t > 0, 0 φ π 2

22 OS Stability p.22/57 Quasi-Schwinger Functions S n ( x 1, x 0 1,, x n, x 0 n) = = lim φ π 2 G n ( x 1, e iφ x 0 1,, x n, e iφ x 0 n) G n S n

23 OS Stability p.23/57 Complex Lorentz Group X := x µ σ µ = ( x 0 + x 3 x 1 ix 2 x 1 + ix 2 x 0 x 3 ) σ µ := (I, σ 1, σ 2, σ 3 ); ( X := x µ ix 0 + x 3 x 1 ix 2 σ eµ = x 1 + ix 2 ix 0 x 3 σ eµ := (ii, σ 1, σ 2, σ 3 ) ).

24 OS Stability p.24/57 Complex Lorentz Group det(x) := (x 0 ) 2 ( x ) 2 det(x) := (x 0 ) 2 ( x ) 2 A B SL(2, C) SL(2, C) X = AXB t X = AXB t det(x) = det(x ) det(x) = det(x )

25 OS Stability p.25/57 Complex Lorentz Group Λ(A, B) µ ν := 1 2 Tr[σ µaσ ν B t ] E(A, B) µ ν := 1 2 Tr[σ eµaσ eν B t ]. (A, B)(A, B ) = (AA, BB )

26 OS Stability p.26/57 Extended Tubes z k = E 1 (A, B)x k a S n (z 1,, z n ) := S n (x 1,, x n ) m D m [E(A, B)] This defines quasi-schwinger functions in a complex Euclidean invariant domain

27 OS Stability p.27/57 Quasi-Wightman Functions quasi-wightman functions can be recovered from quasi-schwinger functions as boundary values of the analytic functions W n ( x 1, x 0 1,, x N, x 0 N ) = lim S n ( x 1, x 0 x 0 1 >x0 2 > >x0 N ix0 1,, x N, x 0 N + ix0 N ) Order of Euclidean times = order of fields in W S n W n

28 OS Stability p.28/57 Reconstruction of QM The reconstruction of an RQM is normally done in terms of quasi-wightman functions, which behave kernels of the physical Hilbert space scalar product. Osterwalder and Schrader found essentially equivalent construction based directly on the quasi-schwinger functions.

29 OS Stability p.29/57 Reconstruction of QM: quasi-wightman x f := {f 0, f 1 (x 11 ), f 2 (x 21, x 22 ),, f k (x k1,, x kk )} f g = d 4 x 1 d 4 x m+n fn(x n,, x 1 ) m,n W m+n (x 1,, x m+n )g m (x n+1,, x n+m ).

30 OS Stability p.30/57 Reconstruction of RQM: quasi-wightman x f x f := Λ 1 (x a) f W m (Λx 1 + a,, Λx m + a) = W m (x 1,, x m ) f g = f g

31 OS Stability p.31/57 Reconstruction of QM - OS x f := {f 0, f 1 (x 11 ), f 2 (x 21, x 22 ),, f k (x k1,, x kk )} θ(x 0, x) := ( x 0, x) x Θf := {f 0, f 1 (θx 11 ), f 2 (θx 21, θx 22 ),, f k (θx k1,, θx kk )}.

32 OS Stability p.32/57 Euclidean form (Θf, Sg) := (f, ΘSg) d 4 x 1 d 4 x m+n fn(θx n,, θx 1 ) m,n S m+n (x 1,, x m+n )g m (x n+1,, x n+m ).

33 OS Stability p.33/57 Reflection Positivity supp x f : t k1 > t k2 > t kk > 0 ( x f S > ) (Θf, Sg) 0

34 OS Stability p.34/57 Physical Hilbert Space f, g S > (Θ(f g), S(f g)) := 0 f g [f] [g] := (Θf, Sg) f [f], g [g]

35 OS Stability p.35/57 Reflection Positivity RP RP must hold subspaces d 4 x 1 d 4 x 4 f2 (θx 2, θx 1 )S 4 (x 1, x 2, x 3, x 4 )f 2 (x 3, x 4 ) 0 d 4 x 1 d 4 x 2 f1 (θx 1)S 2 (x 1, x 2 )f 1 (x 2 ) 0. These conditions constrain G 4 and S 4

36 OS Stability p.36/57 Poincaré Group e i J θ e i P a 3-dimensional rotations 3-dimensional translations e Hx0 positive Euclidean time translations e B θ real rotations in Euclidean space-time planes

37 OS Stability p.37/57 Poincaré Group {0, i f 1 (x 11 ), ( i x 11 x P f := i x 21 )f 2 (x 21, x 22 ), } x 22 ( i x 21 x J f := {0, i x 11 x 11 f 1 (x 11 ), x 21 i x 22 x 22 )f 2 (x 21, x 22 ), }

38 Poincaré Group {0, x 0 11 f 1 (x 11 ), x H f := x x 0 f 2 (x 21, x 22 ), } 22 ( x 21 x 0 x {0, x 11 x 0 11 x 21 + x 22 x B f := x 0 11 x 0 x x 11 f 1 (x 11 ), x 22 )f 2 (x 21, x 22 ), }. OS Stability p.38/57

39 OS Stability p.39/57 Dynamics - Euclidean RQM H, P, J, B well defined and self-adjoint on the Physical Hilbert space H, P, J, B satisfy Poincaré commutation relations, H, P, J, B defines a relativistic quantum theory.

40 OS Stability p.40/57 Euclidean BS G 4 = G 0 + G 0 KG 4 S 4 = S 0 + S 0 K e S 4 G 4 S 4 W 4

41 OS Stability p.41/57 OS Stability S 4 = S 0 + S 0 K e S 4 Π > : S S > (f, Π > ΘS 0 Π > f) 0 K e small, Euclidean covariant? (f, Π > ΘS 4 Π > f) 0

42 OS Stability p.42/57 Cases Case 1: (f, Π > ΘS 0 Π > f) > 0 f 0 Case 2: (f, Π > ΘS 0 Π > f) = 0 for somef 0

43 OS Stability p.43/57 ±t model (f, ΘSf) = (f, 0) Π > := Θ := ( 0 I I 0 ( I ( 0 I I 0 ) ( ) ) s( 0, 0) s( 0, 2t) s( 0, 2t) s( 0, 0) ) ( f 0 = f 2 s( 0, 2t).

44 OS Stability p.44/57 ±t model ( ) ( ) ( s 11 s 21 s 12 s 22 ) ( = s 011 s 021 s 012 s 022 ) ( + ) s 011 s 021 s 012 s 022 k 11 k 21 k 12 k 22 s 11 s 21 s 12 s 22.

45 OS Stability p.45/57 ±t model S 0; Π > ΘSΠ > > 0 k 11 = k 22 < s 11 det(s 0 ) = s 11 s s2 012 k 12 = k 21 > s 012 det(s 0 ) = s 012 s s2 012 k 11 + k 12 < 1 s s 012

46 OS Stability p.46/57 ±t model The inequalities on k ij are satisfied in an open neighborhood containing the origin. This indicates that this trivial model is OS-stable. Parts of the analysis can be generalized more complicated systems. The special property is that Π > ΘSΠ > is bounded from below on S >.

47 OS Stability p.47/57 Null Spaces S 4 = S 0 + S 0 T e S 0 T e = K e + K e S 0 T e. Π > ΘS 4 Π > = Π > ΘS 0 Π > + Π > S 0 ΘT e S 0 Π >

48 OS Stability p.48/57 Null Spaces (Π > f, ΘS 0 Π > f) = 0 (Π > f, S 4 Π > f) = (S 0 Π > f, (ΘT )S 0 Π > f). χ := S 0 Π > f f 0 χ 0 (Π > f, ΘS 4 Π > f) = (χ, ΘT χ)

49 OS Stability p.49/57 Null Spaces (χ, ΘT χ) < 0 instability (χ, ΘT χ) > 0 (χ, Θ( T )χ) < 0 instability

50 OS Stability p.50/57 Null Spaces Invariance Lehmann representation 1 (2π) 2 S 0 (x x ) = d 4 pdm ρ(m)eip0 (t t )+i p ( x x ) (p 0 + iω m ( p ))(p 0 iω m ( p )).

51 OS Stability p.51/57 Null Spaces (f, ΘS 0 f) = dt f(p, t) 2πe ω m( p)t ωm ( p) 2 d 3 pdmρ(m) f(p, t) = 1 (2π) 3/2 e i p x f( x, t)d 3 x

52 OS Stability p.52/57 Null Spaces (f, ΘS 0 f) = 0 dt f(p, t)e ω m( p)t = 0 p and all m supp(ρ).

53 OS Stability p.53/57 Null Spaces S 0 free m is a fixed number f( p, t) = g( p)ξ(t) ˆξ( p) := b a b a ξ(t)dt = 1 supp(ξ) [a, b e ω m(p)t ξ(t)dt f( p, t) = g( p)ξ(t)[1 e ω m(p)t ˆξ( p)] (f, ΘS 0 f) = 0

54 OS Stability p.54/57 Null Spaces For free particles S 0 S 1 S 2 (product of free two-point Schwinger functions). The supports of f = f 1 f 2 can be designed to be in S > and in the null space of Π > ΘS 0 Π > Leads to a violation of the stability condition.

55 Null Spaces If ρ(m) has absolutely continuous spectrum then the instability proof breaks down: f( p, t)e ω m( p)t = 0. n m n dt f(p, t)t n e ω m 2 ( p)t = 0 dt f(p, t)l n (ω m2 ( p)t)e ω m 2 ( p)t = 0 f = 0 OS Stability p.55/57

56 OS Stability p.56/57 Conclusion Euclidean BS with free S 0 is OS unstable Since K e K p, small Minkowski kernels lead to instability. Problem extends to real-time BS equation. The connection of the BS equation with a free S 0 (resp. G 0 ) as driving terms to a relativistic quantum theory is suspect.

57 OS Stability p.57/57 Future What happens in realistic models? Case of Fermions (protons, neutrons, quarks)?