Motion of The Theory and Hamiltonian of The Theory - Physics Assignment Help

Assignment Task

1 Majorana and Dirac Spinors

1. Consider a four-component Majorana spinor 6 = 6., described by the Lagrangian = geoig (0.1) lleat and 6t as independent fields.

(a) Derive the equations of motion of the theory.

(b) Derive the Hamiltonian of the theory.

(c) Calculate the equations of motion again, this time using Hamilton's equations.

(d) What symmetries are explicitly manifest in one approach compared to the other? Explain. 
 

2. In this question we show that the free-field Majorana fermion Lagrangian can be written in terms of a single Weyl fermion 6L, PL. The Majorana and Weyl fields can be related by  6 = 6L + (We, (0.2)

where (Ea = = Cy°*(W* and C is the unitary charge conjugation matrix that satisfies Cy; C-1 = (Unless directed otherwise, answer the following questions without using a specific representation for the -y,` or C.) (a) Explicitly verify that (EL)' = PRE', i.e. that (EL)' is right handed, by choos-ing a representation for C and the e.

(The relation is true independent of representation.) (b) Show that = (c) Show that Cy7 = (d) Use this and the fact that (xc)' = x for a general spinor x to show that C is an antisymmetric matrix, i.e. C = -CT. (e) Use the results of parts (a), (b) and (d) to show that (4)5110,2(6.L)' = roA0A6L+ total divergence. (0.3) You will need to use the fact that 6.6b = -6b6., where a, b are spinor indices. (Alternatively (Ti6' T)T = -6rT6, with some matrix structure.)

This follows because, as we shall see in lectures later on, fermion fields anti-commute. This means that we must introduce a factor of -1 whenever we interchange spinors. (f) Using the results of the previous parts, show that the Lagrangian in Eq. (0.1) can be rewritten into the equivalent form = ieLe0A6L+ 2m(d,C-1&, - GCGT) . 
 

3. How does 6L, transform under spatial parity inversion? 
 

4. Suppose an additional copy of the field G, were added to the theory, so that = 26leat,61, + 2m (612C-16}, - 61,C6})

• (0.5) with i E 11, 2). What continuous symmetry does the Lagrangian now possess? Derive the conserved Noether current J5 and show that the corresponding conserved charge Q is Q = I (em, - 6rd) d3x 

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