B.19 Chapters 43 and 44

B.19 Chapters 43 and 44

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · charge operator

Problem 1:

Show that, assuming the standard Poisson brackets

$$\{\varphi (\mathbf{x}),\pi ({\mathbf{x}}^{\prime })\}=\delta (\mathbf{x}-{\mathbf{x}}^{\prime }),\{\varphi (\mathbf{x}),\varphi ({\mathbf{x}}^{\prime })\}=\{\pi (\mathbf{x}),\pi ({\mathbf{x}}^{\prime })\}=0$$

Hamilton’s equations for the Hamiltonian

$$h={\int }_{{\mathbf{R}}^3}\frac{1}{2}({\pi }^2+(\nabla \varphi {)}^2+m^2{\varphi }^2)d^3\mathbf{x}$$

are equivalent to the Klein-Gordon equation for a classical field $\varphi$

Problem 2:

In section 44.1.2 we studied the theory of a relativistic complex scalar field, with a $U(1)$ symmetry, and found the charge operator $\hat{Q}$ that gives the action of the Lie algebra of $U(1)$ on the state space of this theory.

• Show that the charge operator $\hat{Q}$ has the following commutators with the fields

$$[\hat{Q},\hat{\varphi }]=-\hat{\varphi },[\hat{Q},{\hat{\varphi }}^{†}]={\hat{\varphi }}^{†}$$

and thus that $\hat{\varphi }$ on charge eigenstates reduces the charge eigenvalue by 1, whereas ${\hat{\varphi }}^{†}$ increases the charge eigenvalue by 1.

• Consider the theory of two identical complex free scalar fields, and show that this theory has a $U(2)$ symmetry. Find the four operators that give the Lie algebra action for this symmetry on the state space, in terms of a basis for the Lie algebra of $U(2)$

Note that this is the field content and symmetry of the Higgs sector of the standard model (where the diference is that the theory is not free, but interacting, and has a lowest energy state not invariant under the symmetry).

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