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Faculty of Physics, Lomonosov Moscow State University Advanced Quantum Field Theory: Mo dern Applications in HEP, Astro & Cond-Mat Instructor: O. Kharlanov

Handout 2 (Spring 2015 term)
1. Within the theory of a free massless scalar D = 1 + 1, consider the operator 1 ^ : T00 : (f ) 2 : (t (x, 0))2 + (x (x, 0))2 : f (x)dx, ^ ^

where f (x) is a non-negative infinitely differentiable function with a compact support (a ^ ^ bump function ). Find the norm of the state : T00 : (f ) |0 and the expectation value of A in the state ^ | N (1 + : T00 : (f )) |0 , C, where N is the normalization constant, in terms of the Fourier image of f . Using explicit instances of f (x), show that the density expectation value in | lim ^ | : T00 : (f ) | f (x)dx


suppf {x}

.

is not positive-definite and is unbounded below. 2. Within the problem of an accelerated mirror in a D = 1 + 1 massless scalar theory, start with the integral expression for the regularized stress-energy tensor Tµ (x- ) given in lecture 7 and check the formulae for the renormalized stress-energy T
ren 00

=- T

ren 01

=

1 12

p (x- )

1 p (x- )

.

3. Show that, when the effect of the external magnetic field on the photon dispersion is neglected, the photon splitting to more than 2 photons is kinematically forbidden, namely, the corresponding phase volumes vanish. Moreover, find the double-splitting momentum integral V
(2) ph

(k) =

d3 k1 d3 k2 3 (k1 + k2 - k) (|k1 | + |k2 | - |k|).

4. Using the well-known expression for the Heisenberg­Euler effective action, find the term in the Lagrangian that is responsible for the photon splitting in a strong external magnetic Ї field B (more precisely, for the contribution of the hexagon diagram to the amplitude). 5. Within a scalar D = 1 + 1 theory with the Lagrangian L= 1 (µ )2 - (2 - v 2 )2 , 2 4 m v ,

find the mass counterterm L = - 1 m2 2 which renormalizes the self-energy, within the 2 leading order in . In the integrals, use the momentum cutoff |p| < and explicitly expand the counterterm in divergent powers of . 6. Prove the Fierz identities (, , , are classical spinor fields) L µL = L µL , Їµ Ї Їµ Ї where P
L,R

Ї R µL = -2PL PL , Їµ Ї Ї

(1

µ 5 )/2, L,R µ PL,R .

7. Starting with the general expression for the 1-loop effective potential for axions in the AWZ model (see lecture 11), use the momentum cutoff regularization |p| < and find the divergent and finite parts of the potential.

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