A Model of Quantum Field Theory with a Fundamental Length S. - PowerPoint PPT Presentation

1 A Model of Quantum Field Theory with a Fundamental Length S. Nagamachi The University of Tokushima 1. Introduction 2. Wightman Axioms 3. Fundamental Length 4. Ultrahyperfunction 5. Model 6. Continuous Limit

2 1 Introduction The relativistic equation of quantum mechanics called Dirac equation ∂ i � c γ µ ψ ( x ) − Mψ ( x ) = 0 , x 0 = ct, x 1 = x, x 2 = y, x 3 = z ∂x µ contains the constants: c (velocity of light): the fundamental constant in the relativity theory, h = 2 π � (Planck constant): the fundamental constant in quantum mechanics. Dimension: c : [LT − 1 ], h : [ML 2 T − 1 ]. W. Heisenberg thought that the equation must also contain a constant l with dimension [L]. Arbitrary dimensions are expressed by the combination of c , h and l , e.g., [T] = [L]/[LT − 1 ], [M] = [ML 2 T − 1 ]/([LT − 1 ][L])

3 In 1958, Heisenberg with Pauli introduced the equation ∂ � ψ ( x ) ± l 2 γ µ γ 5 ψ ( x ) ¯ ψ ( x ) γ µ γ 5 ψ ( x ) = 0 , c γ µ (1) ∂x µ which is later called the equation of universe. The constant l has the dimension [L] and is called the fundamental length. D¨ urr, H.-P.; Heisenberg, W.; Mitter, H.; Schlieder, S.; Yamazaki, K. Zur Theorie der Elementarteilchen, Z. Naturf. 14a (1959) 441-485 Heisenberg, W., Introduction to the Unified Field Theory of Elementary Particles, John Wiley & Sons (1966) 1965 Shin’ichiro Tomonaga was awarded the Nobel prize for physics. 1967 Heisenberg visited to Japan for the second time (first time 1929). Heisenberg gave a talk in Kyoto University.

4 But equation (1) is difficult to solve. So, we consider the following soluble equation having the constant l with the dimension [L]: � cm � 2  � φ ( x ) + φ ( x ) = 0   � (2) . � ∂ � ψ ( x ) = 2 γ µ l 2 ψ ( x ) φ ( x ) ∂φ ( x ) i � c γ µ − M   ∂x µ ∂x µ This equation has no solutions in the axiomatic framework of of Wightman, that is, the field ψ ( x ) is not an operator-valued tempered distribution. But ψ ( x ) is an operator-valued tempered ultrahyperfunction. The equation (2) has a solution in the framework of E. Br¨ uning and S. Nagamachi: Relativistic quantum field theory with a fundamental length, J. Math. Phys. 45 (2004) 2199-2231.

5 2 Wightman axioms W.I (Relativistic invariance of the state space). There is a physical Hilbert space H in which a unitary representation U ( a, A ) of the Poinar´ e spinor group P 0 acts. W.II (Spectral property). W.III (Existence and uniqueness of the vacuum). There exists in H a unique unit vector Ψ 0 (called the vacuum vector), W.IV (Fields and temperedness). The components φ ( κ ) of the quantum field φ ( κ ) j are operator-valued generalized functions φ ( κ ) ( x ) over the Schwartz space S ( R 4 ) j with common dense domain of definition D to all the operaotrs φ ( κ ) ( f ) . j W.V (Cyclicity of the vacuum). W.VI (Poincar´ e-covariance of the fields).

6 W.VII (Locality, or microcausality). ( x ) and φ ( κ ′ ) Any two field components φ ( κ ) ( y ) either commute or anti-commute j ℓ under a spacelike separation of x and y : If f and g have space-like separeted supports ( f ) φ ( κ ′ ) ( g )Ψ ∓ φ ( κ ′ ) φ ( κ ) ( g ) φ ( κ ) ( f )Ψ = 0 j j ℓ ℓ for all Ψ ∈ D . We express ( x )Ψ = 0 for ( x − y ) 2 < 0 ( x ) φ ( κ ′ ) ( y )Ψ ∓ φ ( κ ′ ) φ ( κ ) ( y ) φ ( κ ) j ℓ ℓ j [( x − y ) 2 = ( x 0 − y 0 ) 2 − ( x 1 − y 1 ) 2 − ( x 2 − y 2 ) 2 − ( x 3 − y 3 ) 2 ]

7 3 Fundamental length W.VII (Locality) says that the two events which are space-likely separated are independent. Even if we replace W.VII by a weaker axiom ( x )Ψ = 0 for ( x − y ) 2 < − ℓ 2 < 0 , ( x ) φ ( κ ′ ) ( y )Ψ ∓ φ ( κ ′ ) φ ( κ ) ( y ) φ ( κ ) j j ℓ ℓ (the two events which are separated by ℓ are independent), we can prove W.VI ( x )Ψ = 0 for ( x − y ) 2 < 0 ( x ) φ ( κ ′ ) ( y )Ψ ∓ φ ( κ ′ ) φ ( κ ) ( y ) φ ( κ ) j j ℓ ℓ by using other axioms. It is not easy to weaken the condition of locality if the field φ ( κ ) ( x ) has the localization property. We must introduce generalized functions j which have no localization property.

8 Let T ( − ℓ, ℓ ) = R + i ( − ℓ, ℓ ) ⊂ C . T ( T ( − ℓ, ℓ )) ∋ f : holomorphic function in T ( − ℓ, ℓ ) . Then for | a | < ℓ , we have � ∞ ∞ ∞ a n ( − a ) n � � n ! δ ( n ) ( x ) f ( x ) dx = f ( n ) (0) n ! −∞ n =0 n =0 � ∞ = f ( − a ) = δ ( x + a ) f ( x ) dx. −∞ (A): ∆ N ( x ) = � N a n n ! δ ( n ) ( x ) converges to δ ( x + a ) = δ − a ( x ) in T ( T ( − ℓ, ℓ )) ′ n =0 as N → ∞ . supp ∆ N = { 0 } , supp δ − a = {− a } . (B): If | a | > ℓ , ∆ N ( x ) does not converge in T ( T ( − ℓ, ℓ )) ′ . (A) and (B) imply: If | a | < ℓ then the distinction between { 0 } and {− a } is not clear in T ( ℓ ) ′ , but if | a | > ℓ then the distinction between { 0 } and {− a } is clear.

9 4 Ultrahyperfunction Hasumi, M., Tohoku Math. J. 13 (1961) Morimoto, M., Proc. Japan Acad. 51 (1975) T ( A ) = R n + iA ⊂ C n , A ⊂ R n . R n ⊃ K : convex compact T b ( T ( K )) ∋ f : f is continuous on T ( K ) , holomorphic in the interior of T ( K ) and satisfy � f � T ( K ) ,j = sup {| z p f ( z ) | ; z ∈ T ( K ) , | p | ≤ j } < ∞ , j = 0 , 1 , . . . . There is a natural mapping for K 1 ⊂ K 2 T b ( T ( K 2 )) → T b ( T ( K 1 )) .

10 Let O be a convex open set in R n . We define T ( T ( O )) = lim ← T b ( T ( K )) , K ↑ O. T ( T ( O )) : Fr´ echet space Definition 4.1 tempered ultrahyperfunction is a linear form on the space T ( T ( R n )) . T ( T ( R n )) ′ : space of tempered ultrahyperfunctions In the book of I.M. Gel’fand and G.E. Shilov, Generalized functions Vol. 2, (1968), there are function spaces S 1 ,B and S 1 = lim B →∞ S 1 ,B = K 1 →{ 0 } T b ( T ( K 1 )) , lim 0 ← B S 1 ,B = T b ( T ( K 1 )) = T ( T ( R n )) . but no space lim lim R n ← K 1

11 5 Model Lagrangian density: Natural unit, c = � = 1 . L ( x ) = L F f ( x ) + L F b ( x ) + L I ( x ) , L F f ( x ) = ¯ ψ ( x )( iγ µ ∂ µ − ˜ m ) ψ ( x ) , L F b ( x ) = 1 2 { ( ∂ µ φ ( x )) 2 − m 2 φ ( x ) 2 } , L I ( x ) = 2 l 2 ( ¯ ψ ( x ) γ µ ψ ( x )) φ ( x ) ∂ µ φ ( x ) . The field equations  ( � + m 2 ) φ ( x ) = 0  � � ∂ ψ ( x ) = 2 γ µ l 2 ψ ( x ) φ ( x ) ∂φ ( x ) iγ µ − ˜ m ∂x µ ∂x µ 

12 Quantization – Path integral. Two point function, formally �� � � ¯ d D ( ψ, ¯ ψ α ( x 1 ) ψ β ( x 2 ) exp i R 4 L I ( x ) dx ψ ) d G ( φ ) � − 1 �� �� � d D ( ψ, ¯ × exp i R 4 L I ( x ) dx ψ ) d G ( φ ) , �� � � d G ( φ ) = exp i R 4 L F b ( x ) dx dφ ( x ) x ∈ R 4 4 �� � � d D ( ψ, ¯ ψ α ( x ) ¯ � ψ ) = exp i R 4 L F f ( x ) dx ψ α ( x ) . α =1 x ∈ R 4 Lattice approximation. M, N : positive integers L = MN . Γ = { t = j ∆; j ∈ Z , − L < j ≤ L, ∆ = √ π/M } = ∆ Z / (2 √ πN ) .

13 Linear operator −△ + m 2 on R Γ 4 = R 4 · 2 L (difference operator on the lattice Γ 4 ) 3 −△ + m 2 : R Γ 4 ∋ Φ( x ) → − Φ( x + e µ ) + Φ( x − e µ ) − 2Φ( x ) + m 2 Φ( x ) ∈ R Γ 4 . � ∆ 2 µ =0 Gaussian measure on R 4 · 2 L : � 3  1 Φ( y + e µ ) + Φ( y − e µ ) − 2Φ( y )  � � dG (Φ) = C exp ∆ 2 2  µ =0 y ∈ Γ 4 ∆ 4 � � − m 2 Φ( y ) � d Φ( y ) , y ∈ Γ 4 � dG (Φ) = 1 . The exponent: Euclideanized ( x 0 → C : normalization constant − iy 0 , x → y ) discretization of Lagrangian i � L F b ( x ) dx .

14 The covariance � Φ( y 1 )Φ( y 2 ) dG (Φ) = 2( −△ + m ) − 1 ( y 1 , y 2 ) = 2 S m ( y 1 − y 2 ) � 3 � − 1 (2 − 2 cos p µ ∆) / ∆ 2 + m 2 S m ( y 1 − y 2 ) = (2 π ) − 4 � � e ip ( y 1 − y 2 ) η 4 , p ∈ ˜ µ =0 Γ 4 Γ = { s = jη ; j ∈ Z , − L < j ≤ L, η = √ π/N } = η Z / (2 √ πM ) . ˜ Nonstandard analysis: S m ( y 1 − y 2 ) → S m ( y 1 − y 2 ) , M, N → ∞ . Schwinger function of neutral scalar field of mass m : � p 2 + m 2 � − 1 d 4 p. S m ( y 1 − y 2 ) = (2 π ) − 4 R 4 e ip ( y 1 − y 2 ) �

15 Measure dD (Ψ 1 , Ψ 2 ) on the Grassmann algebra generated by { Ψ 1 α ( y ) , Ψ 2 α ( y ); α = 1 , . . . , 4 , y ∈ Γ 4 } : � 3   �   dD (Ψ 1 , Ψ 2 ) = C ′ exp Ψ 2 T ( y ) � � γ E Ψ 1 ( y )∆ 4  − µ ∇ µ + ˜ m  µ =0 y ∈ Γ 4 4 � � d Ψ 1 α ( y ) d Ψ 2 × α ( y ) , α =1 y ∈ Γ 4 Ψ 1 = (Ψ 1 4 ) T , Ψ 2 = (Ψ 2 4 ) T , 1 , . . . , Ψ 1 1 , . . . , Ψ 2 � σ 0 � � � 0 0 − iσ j γ E , γ E 0 = γ 0 = j = − iγ j = , j = 1 , 2 , 3 , 0 − σ 0 iσ j 0 � 1 � 0 � 0 � 1 � � � � 0 1 − i 0 σ 0 = , σ 1 = , σ 2 = , σ 3 = , 0 1 1 0 i 0 0 − 1

16 � ∇ + Ψ k ( y ) = (Ψ k ( y + e µ ) − Ψ k ( y )) / ∆ if k = 1 , 2 , ∇ µ Ψ k = ∇ − Ψ k ( y ) = (Ψ k ( y ) − Ψ k ( y − e µ )) / ∆ if k = 3 , 4 . Avoid doubling problem. 3 − L I ( y ) = Ψ 2 T ( y ) e − il 2 Φ( y ) 2 � γ E µ µ =0 × [ P + Ψ 1 ( y + e µ ) { e − il 2 Φ( y + e µ ) 2 − e − il 2 Φ( y ) 2 } / ∆ + P − Ψ 1 ( y − e µ ) { e − il 2 Φ( y ) 2 − e − il 2 Φ( y − e µ ) 2 } / ∆] , P ± = (1 ± γ E 0 ) / 2 . L I ( y ) → iL I ( x ) : differences → derivatives, ( y 0 → ix 0 , y → x ).