Continuous symmetries and conserved currents based on S-22 Consider a set of scalar fields, and a lagrangian density let s make an infinitesimal change: variation of the action: setting we would get equations of motion 140 thus we find: this is called Noether current; now we have: = 0 if eqs. of motion are satisfied if a set of infinitesimal transformations leaves the lagrangian unchanged, invariant,, the Noether current is conserved! charge density current density 141
we set Dimensions any quantity A has units of mass to some power that we call [A], e.g: it allows us to convert a time T to a lenght L: a length to an inverse mass: the action appears in the exponential and so in d spacetime dimensions and for the lagrangian density we have: 142 from the kinetic term: in 4 dimensions: functional derivative: dim = f dim = 1 dim = -f+1 143
similarly, in 4d: dim = 1 dim = 4 dim = 3 Noether current: dim = 4 dim = 1 dim = 3 dim = 3 dim = 1 and so on... 144 Consider a theory of a complex scalar field: clearly is left invariant by: in terms of two real scalar fields we get: U(1) transformation (transformation by a unitary 1x1 matrix) and the U(1) transformation above is equivalent to: SO(2) transformation (transformation by an orthogonal 2x2 matrix with determinant = +1) 145
infinitesimal form of is: and the current is: we treat and as independent fields 146 repeating the same for the SO(2) transformation: the Noether current is: which is equivalent to 147
Let s define the Noether charge: we find: integrating over, using Gauss s law to write the volume integral of as a surface integral and assuming on that surface Q is constant in time! using free field expansions, we get: for an interacting theory these formulas are valid at any given time counts the number of a particles minus the number of b particles; it is time independent and so the scattering amplitudes do not change the value of Q; in Feynman diagrams Q is conserved in every vertex. 148 Another use of Noether current: Consider a transformation of fields that change the lagrangian density by a total divergence: there is still a conserved current: e.g. space-time translations: we get: stress-energy or energy-momentum tensor 149
for a theory of a set of real scalar fields: we get: in particular: hamiltonian density then by Lorentz symmetry the momentum density must be: plugging in the field expansions, we get: as expected 150 The energy-momentum four-vector is: Recall, we defined the space-time translation operator so that we can easily verify it; for an infinitesimal transformation it becomes: it is straightforward to verify this by using the canonical commutation relations for and. 151
The same procedure can be repeated for Lorentz transformations: the resulting conserved current is: antisymmetric in the last two indices as a result of being antisymmetric the conserved charges associated with this current are: again, one can check all the commutators... generators of the Lorentz group 152 Recall from S-2: Discrete symmetries: P, T, C and Z Infinitesimal Lorentz transformation: not all LT can be obtained by compounding ILTs! based on S-23 +1 proper -1 improper proper LTs form a subgroup of Lorentz group; ILTs are proper! Another subgroup - orthochronous LTs, ILTs are orthochronous! REVIEW 153
When we say theory is Lorentz invariant we mean it is invariant under proper orthochronous subgroup only (those that can be obtained by compounding ILTs) Transformations that take us out of proper orthochronous subgroup are parity and time reversal: orthochronous but improper nonorthochronous and improper REVIEW A quantum field theory doesn t have to be invariant under P or T. 154 For every proper orthochronous LT there is a unitary operator: we expect the same for parity and time-reversal so that scalar (even under parity) since and we need: that can be also satisfied with: pseudoscalar (odd under parity) 155
We can choose the transformation properties of fields. It is a part of specifying the theory. But if possible we want to have lagrangian density even under both parity and time-reversal, so that parity and time-reversal are conserved. Note: time-reversal operator must be antiunitary: to see it, let s look at transformations of the energy-momentum 4-vector: can be checked directly using: the same result for scalar and pseudoscalar 156 for we have: GOOD hamiltonian is invariant under both parity and time reversal if T was unitary, we would have which is a DISASTER since hamiltonian is invariant under time-reversal only if and so. Let s trace the origin of antiunitarity: the spacetime translation operator implies: 157
for an infinitesimal translation we get: similarly for time-reversal: comparing linear terms in a we see that in order to get we need T is antiunitary 158 symmetry: we want to consider a possibility that the sign of a scalar field changes under a symmetry transformation (that does not act on spacetime arguments). The corresponding unitary operator is: e.g. a theory of a complex scalar field: symmetry has U(1) symmetry: but also an additional discrete symmetry:, equivalent to SO(2): charge conjugation 159
Charge conjugation always occurs as a companion to a U(1) symmetry; it enlarges SO(2) symmetry (the group of 2x2 orthogonal matrices with determinant +1) into O(2) symmetry (the group of 2x2 orthogonal matrices) We can define the operator of charge conjugation: and the charge conjugation is a symmetry of the theory: or Scattering amplitudes must be unchanged if we exchange all a-type particles (charge +1) with all b-type particles (charge -1). This is only possible if both particles have the same mass; we say particle b is antiparticle of a. 160 Another example of consider theory: symmetry: this theory is obviously invariant under: the ground state (if unique) must also be an eigenstate of Z; we can fix the phase of Z via: and then we have: any choice would be fine the symmetry implies that there is no need for a counterterm! 161
Nonabelian symmetries Let s generalize the theory of two real scalar fields: based on S-24 to the case of N real scalar fields: the lagrangian is clearly invariant under the SO(N) transformation: orthogonal matrix with det = 1 lagrangian has also the symmetry,, that enlarges SO(N) to O(N) 162 infinitesimal SO(N) transformation: antisymmetric R T ij = δ ij + θ ji R 1 ij = δ ij θ ij Im(R 1 R) ij = Im R ki R kj = 0 k (N^2 linear combinations of Im parts = 0) there are linearly independent real antisymmetric matrices, and we can write: or R = e iθa T a. The commutator of two generators is a lin. comb. of generators: real hermitian, antisymmetric, NxN generator matrices of SO(N) we choose normalization: structure constants of the SO(N) group 163
e.g. SO(3): Levi-Civita symbol 164 consider now a theory of N complex scalar fields: the lagrangian is clearly invariant under the U(N) transformation: group of unitary NxN matrices we can always write so that. actually, the lagrangian has larger symmetry, SO(2N): SU(N) - group of special unitary NxN matrices U(N) = U(1) x SU(N) 165
infinitesimal SU(N) transformation: hermitian or Ũ = e iθa T a. traceless there are linearly independent traceless hermitian matrices: e.g. SU(2) - 3 Pauli matrices SU(3) - 8 Gell-Mann matrices the structure coefficients are, the same as for SO(3) 166