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note11 : October 8, 2012 CPT theorem Combined transformation: CPT The CPT theorem states that in a canonical quan- Under the combination of all three the field φ tum field theory the action is under changes the sign of the argument, the combination of charge conjugation (C), parity CP T transformation (P ), and reversal (T ). φ(x) φ( x) . (7) The CPT theorem entails certain relations be- −→ − tween physical observables, in particular equal However, since the action involves integration d4x and equal life- for particles and an- over the whole space, the sign change of theR argu- tiparticles. At present, CPT is the sole combina- ment of a field leaves the action unchanged, q.e.d. tion of C, P , T observed as an exact of nature at the fundamental level. Gauge theories Let us see how it works using the complex field as an example. is a peculiar quantum field theory where the Lagrangian is invariant under continuous C: charge conjugation of local transformations, called gauge trans- formations. The transformations form a Lie group Under charge conjugation the particles are ex- which is referred to as the symmetry group or gauge changed with , group of the theory. For each group parameter there is a special vec- C ak bk . (1) tor field, called gauge field, which ensures the in- −→ variance of the Lagrangian under the gauge trans- † The field φ(x) then turns into φ (x), formation. The quanta of the gauge field are called gauge bosons. 1 −ikx † +ikx C φ(x)= ake + bke If the symmetry group is commutative, the gauge √2ωk −→ Xk   theory is called abelian, otherwise it is called non- 1 −ikx † +ikx † abelian or Yang-Mills theory. bke + ake = φ (x) . (2) √ ωk Historically the quantum electro-dynamics Xk 2   (QED) was first recognised as a gauge theory with the gauge group U(1). Then it turned out P: parity transformation that only gauge theories can be renormalized and Under parity transformation the therefore the had to be built as a changes sign, gauge theory, though in order to incorporate more P ak a−k . (3) gauge bosons the gauge group had to be enlarged. → It turned out that the groups SU(2) and SU(3) The field φ(t, x) then turns into φ(t, x), − fit perfectly for the weak and strong interactions correspondingly. 1 −ikx † +ikx P φ(x)= ake + bke k √2ωk   → X Quantum electro-dynamics 1 −ikx † +ikx a−ke + b−ke = φ(t, x) . (4) (QED) is a theory of the k √2ωk   − X /positron () field ψ coupled to the electromagnetic (vector) field Aa with the interac- T: time reversal a tion Lagrangian jaA , where ja = gψγ¯ aψ is the − Under time reversal the process of annihilation of conserved current and g is the charge of electron. a particle turns into the process of generation of a The QED Lagrangian, particle with opposite momentum, a 1 ab QED = ψγ¯ i(∂a +igAa)ψ mψψ¯ FabF , (8) T † L − − 4 ak a k . (5) → − is invariant under the local gauge transformation, The field φ(t, x) then turns into φ†( t, x), − ψ ψ′ = eigα(x)ψ → . (9) 1 −ikx † +ikx T A A′ = A ∂ α(x) φ(x)= ake + bke  a a a a √ ωk → − Xk 2   → where α(x) is an arbitrary scalar function (and also 1 † −ikx +ikx † a−ke + b−ke = φ ( t, x) . (6) the group parameter). The transformation matri- √2ωk − Xk   ces eigα form a Lie group U(1) where the charge

1 note11 : October 8, 2012 is the group generator. QED is thus a gauge the- an object rotated by a (usually fundamental) rep- ory with the symmetry group U(1). The group is resentation of the gauge group. commutative, therefore QED is an abelian theory. The Yang-Mills guage-field is The QED Lagrangian (8) can be conveniently 1 written as1 F = [D ,D ] ab ig a b a 1 2 QED = iψγ¯ Daψ mψψ¯ F , (10) = ∂aAb ∂bAa + ig[Aa, Ab] L − − 4 j − = Ij Fab . (17) where the group covariant derivative, The Yang-Mills Lagrangian for the gauge field is Da = ∂a + igAa , (11) 1 2 1 ab j YM = Trace(F )= Fj F . (18) has the property that under the gauge transforma- L −2 −4 ab tion igα(x) with the standard normalization Daψ e Daψ . (12) → I I 1 δ . The gauge field tensor can be written through Trace( j k)= 2 jk (19) group covariant derivatives as Since the Yang-Mills field tensor contains the 1 gauge field (which is of second order in F = [D ,D ] . (13) ab ig a b the field) the Yang-Mills Lagrangian contains third and fourth order terms. The non-abelian gauge Non-abelian (Yang-Mills) gauge theories bosons can thus self-interact. The gauge bosons are necessarily massless (as the The QED with the U(1) symmetry group has one term breaks gauge invariance). However with gauge boson, the , while more gauge bosons the so called the gauge bosons can are needed for the standard model. Yang and Mills acquire effective mass through interactions with the suggested to consider a more general group of ma- Higgs field. trices, The number of gauge bosons is equal to the num- U(x)= eiIj αj (x) (14) ber of generators in the gauge group. There is one gauge boson, the photon, for the U(1) group; three with generators Ij , j =1 ...n, and Lie-algebra gauge bosons for the SU(2) group; and eight gauge l bosons for the SU(3) group. [Ij , Ik]= CjkIl. (15)

l If the structure constants Cjk are not equal zero, the theory is called non-abelian. To make the theory gauge invariant a separate a vector field Aj is needed for each group parameter a j a αj and the gauge field becomes A = I Aj (there is no difference whether the group index j is up or down). The generalized gauge transformation is now defined as ψ ψ′ = Uψ , → ′ −1 1 −1  Aa Aa = UAaU (∂aU)U . → − ig (16) The transformation matrices U are of a certain and thus the field ψ, on which the ma- trix operates, becomes a column-vector in this new dimension2. In a Yang-Mills theory the fermionic field ψ is then a rather non-trivial object: it is a genera- tion/annihilation in the space of quantum states of the field; a bispinor; and

1 2 ab F ≡ F Fab 2this dimension is referred to as weak isospin space for the and color space for the .

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