8.324 Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 2010 Lecture 3

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1 Lecture Relativistic Quantum Field Theory II Fall Relativistic Quantum Field Theory II MIT OpenCourseWare Lecture Notes Hong Liu, Fall 200 Lecture 3 We begin with some comments concerning gauge-symmetric theories:. A U() local symmetry leads to the field A µ (x), mediating interactions between the charge fields ψ i (x). 2. No mass term is allowed for A µ or the gauge symmetry is broken. 3. It is possible to construct theories using other gauge invariant terms, for example ϵ µνλρ F µν F λρ, (F µν F µν ) 2, ψf µν µ ν ψ. () All of these terms can be written as a total derivative, or are non-renormalizable. 4. Some of the terminology and constructs associated with gauge theories includes: U(y, x) : Parallel transport, A µ (x) : Connection, F µν (x) : Curvature, D µ (x) : Covariant derivative. These objects form the mathematical framework of the fibre bundle. 5. Gauge symmetry is not really a symmetry. The phase of ψ (and part of A µ ) does not carry physical information; there is a redundancy in our variables. x n y x 2 x x Figure : Dividing the path into infinitesimal segments. 6. Finally, we consider the explicit form of the finite parallel transport, U(y, x): choose a path from x y. We split the path into infinitesimal segments: U (y, x) = U(y, x n )U(x n, x n )... U(x, x), (2) where, from our result from the previous lecture for the infinitesimal parallel transport, U(x, x) exp [iea µ (x x) µ ], (3) and hence [ ˆ ] U (y, x) = exp ie A µ (x)dx µ. (4) Note that U (y, x) is not necessarily path-independent: in general, U (y, x) = U 2 (y, x). Let U Γ (x, x) = U 2 (y, x)u (y, x) (5) be the parallel transport associated with the closed loop shown in figure 2. By (4), [ ] U Γ (x, x) = exp ie A µ dx µ, (Γ = 2 ). (6) Γ Using Stoke s theorem, we will see in the problem set that [ ˆ ] U Γ (x, x) = exp ie F µν dx µ dx ν. (7) Σ U Γ (x) is the Wilson loop, and it is associated with phenomena such as the Aharonov-Bohm effect, and Berry s phase.

2 Lecture Relativistic Quantum Field Theory II Fall 200 y 2 Σ x Figure 2: Paths and 2 from x to y, and the enclosed area Σ..3: NON-ABELIAN GENERALIZATIONS: YANG-MILLS THEORY ψ Now consider Ψ =. ψ n and search for a theory invariant under Ψ(x) Ψ (x) = V (x)ψ(x), (8) or, with indices restored, Ψ i (x) Ψ i(x) = V j i (x)ψ j (x). (9) Here, V (x) is an n n unitary matrix of unit determinant, that is, V (x) SU(n). Let us construct the non-abelian generalizations of the objects we studied in the Abelian case. A. Covariant derivative: Introduce U(y, x) SU(n), transforming as Again, for y µ = x µ + ϵn µ, taking the limit ϵ 0, we expand U(x + ϵn, x) : U(y, x) V (y)u(y, x)v (x). (0) U(x + ϵn, x) = + igϵn µ A µ (x) +..., () where g is a constant, and A µ (x) is an n n matrix. As U(y, x) SU(n), A µ (x) is also necessarily traceless and hermitian; A µ (x) = A µ(x). Inserting this expansion into the transformation law (0), we obtain the gauge transformation law for the connection: As before, we define the covariant derivative by i A µ (x) V (x)a µ (x)v (x) g ( µ V (x))v (x). (2) n µ D µ Ψ lim [Ψ(x + ϵn) U(x + ϵn, x)ψ(x)]. (3) ϵ 0 ϵ Hence, we have, with indices written explicitly, (D µ Ψ) i = µ Ψ i ig(a µ ) j Ψ j. (4) i From (8) and (2), we have that, under a gauge transformation, D µ Ψ(x) V (x)(d µ Ψ)(x), and so, the Lagrangian L = iψ( µ D µ m)ψ is invariant. B. Kinetic term for A µ : We note that under a gauge tranformation, and that [D µ, D ν ] Ψ V [D µ, D ν ] Ψ, (5) [D µ, D ν ] = [ µ iga µ, ν iga ν ] = ig( µ A ν ν A µ ig [A µ, A ν ]) igf µν, (6) 2

3 Lecture Relativistic Quantum Field Theory II Fall 200 with F µν an n n matrix satisfying F µν that and, as Ψ = V Ψ, we have that = F µν and TrF µν = 0. Under a gauge transformation, from (8), we have F µν Ψ F µν Ψ = V F µν Ψ, (7) F µν (x) = V (x)f µν (x)v (x), (8) and so F µν is gauge covariant, as can be checked directly from (2). Hence, Tr(F µν F µν ) is invariant under gauge transformations. C. The Lagrangian: We can now write down an invariant L : L = c Tr(F µν F µν ) i Ψ( µ D µ m)ψ, (9) 4 ψ with Ψ =., D µ = µ iga µ, F µν = µ A ν ν A µ ig [A µ, A ν ], and A µ = A µ, TrA µ = 0. Explicitly, this ψ n Lagrangian is invariant under the local gauge transformation: Ψ(x) Ψ (x) = V (x)ψ(x), i A µ (x) A µ (x) = V (x)a µ (x)v (x) ( µ V (x))v (x), g F µν (x) = V (x)f µν (x)v µν (x) F (x), where V (x) = exp [iλ a (x)t a ], and T a are the generators of the Lie algebra, satisfying [T a, T b ] = if abc T c. Remarks:. A µ is massless; introducing a mass term breaks gauge invariance. 2. It is convenient to expand A µ as A µ = A a µ T a, with a =, 2, n 2. Here, A µ is an n n matrix, and A a µ are n 2 ordinary functions of x. Similarly, F = F a µν µν T a. We have that F a = µ A a ν T a ν A a T a ig [ A b T b, A c ν T c ], (20) µν T a µ µ and so, explicitly, F a A a a b µν = A c µ ν ν A µ + gf abc A µ ν. (2) 3. There is another gauge invariant term which can be constructed out of F at the quadratic level: ϵ µνλρ Tr(F µν F λρ ). (22) However, this term breaks CP-symmetry, and is also a total derivative. Nevertheless, this term is important at a non-perturbative level, as we will see in Non-Abelian gauge fields are associated with fibre bundles..3.2: The Wilson loop U (y, x) = lim U(y, x n )U(x n, x n )... U(x, x) (23) n n = lim ( + iga µ (x j ) x j µ ), x j 0 j=0 3

4 Lecture Relativistic Quantum Field Theory II Fall 200 µ µ with x j = x j + x µ j, x 0 = x, x n+ = y. Note that the ordering is important in (23), as A µ is a matrix, and so [A µ (x j ), A ν (x k )] = 0 in general. U (y, x) = + ig n j=0 Now, we introduce x µ (s) to parameterise : Then A µ (x j ) x µ j + (ig)2 n j A µ (x j ) x µ j A v (x k ) x v k (24) j=0 k=0 x µ (0) = 0, x µ () = y µ, s [0, ]. (25) ˆ ˆ ˆs ds A µ (x(s )) dxµ + (ig) 2 dx µ dx v U (y, x) = + ig ds ds 2 A µ (x(s )) A v (x(s 2 )) (26) ds ds ds ˆ ˆs sˆ n = dx µ dx µ n (ig) n ds ds 2... ds n A µ (x(s ))... A µn (x(s n )) (27) ds n=0 ds n ˆ dx ig P exp ds A µ (x(s)) (28) ds 0 ˆ =P exp ig A µ (x)dx µ. (29) By construction, under a gauge transformation, U (y, x) V (y)u (y, x)v (x). (30) To prove this directly using (2) is slightly non-trivial. As in the Abelian case, in general U (y, x) = U 2 (y, x). For a closed loop Γ, U Γ (x, x) is nontrivial. The non-abelian generalisation of Stokes theorem can be used to relate the parallel transport around the loop to the flux passing through the loop. For an infinitesimal loop, U Γ (x, x)ψ Ψ = 2 F µν σ µν Ψ, (3) where σ µν is the area element encircled by the loop. Under a gauge transformation, U Γ (x, x) V (x)u Γ (x, x)v (x), (32) and hence, W Γ (x) = Tr(U Γ (x, x)) = Tr(P exp ig A µ dx µ ) (33) Γ is gauge invariant. This is the non-abelian Wilson loop. It is a very important object. 4

5 MIT OpenCourseWare Relativistic Quantum Field Theory II Fall 200 For information about citing these materials or our Terms of Use, visit: