MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
7
3. Feynman calculus
3.1. Wicks theorem. Let V be a real vector space of dimension d with volume element dx. Let S (x)
be a smooth function on a box B V which attains a minimum at x = c Interior(B ),
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MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
8. Operator approach to quantum mechanics
In mechanics and eld theory (both classical and quantum), there are two main languages Lagrangian and Hamiltonian. In the classical setting, the Lagrangian
Homework 4; due Thursday, Oct. 17
1. Prove Mumfords theorem (see the notes, Th. 5.2) for g = 1.
2. Let (N ) be the congruence subgroup of S L2 (Z) which consists of matrices
equal to 1 modulo N .
(a) Show that (N ) is free for N 3. (Hint: consider the act
Homework 7; due Thursday, Dec. 5
1. Consider quantum mechanics with Yukawa coupling. That is, we have a scalar
boson (t) and two fermions 1 (t), 2 (t), and the Lagragian is
1
L = (2 m2 2 + 1 1 + 2 2 1 2 ) + g 1 2 .
2
Compute the 2-point function < (t)(0)
Homework 5; due Tuesday, Nov. 5
1. Calculate the 1-particle irreducible 2-point function for a quantum particle
with potential U (q ) = m2 q 2 /2 + gq 4 /4! modulo g 3 (in momentum space, for = 1).
In class we did it modulo g 2 .
2. Let U (q ) = m2 q 2 /2
Homework 2; due Tuesday, Sept. 24
1. Compute the 1-loop contribution to ln(Z/Z0 ) for S (x) = x2 /2 g (x + x3 /6).
Using this, compute the number of labeled n-vertex 1-loop graphs with 1-valent
and 3-valent vertices only.
2. Find the generating function
a
Homework 6; due Thursday, Nov. 21
1. Let A be a block 2 by 2 matrix over a supercommutative ring R with A11 , A22
being square matrices of sizes n 0, m 0 with even entries, and A12 , A21 having
odd entries. The supertrace of A is strA := trA11 trA22
(a) S
Homework 1; due Tuesday, Sept. 17
1. Write a complete proof of Theorem 1.1. (i.e. ll the gaps left in the lecture
notes)
2* (slightly harder). Prove Theorem 1.2.
3. Calculate 0 sinn (x)dx for nonnegative integers n, using integration by parts.
Then apply
Homework 3; due Thursday, Oct. 3
1. Generalize tHoofts theorem to integrals over quaternionic Hermitian
matrices.
2. Find the number of ways to glue an orientable surface of genus g 1 from a
4g -gon (the gluing must preserve orientation), and prove your a
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MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
9. Fermionic integrals
9.1. Bosons and fermions. In physics there exist two kinds of particles bosons and fermions. So
far we have dealt with bosons only, but many important particles are fermions:
MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
37
7. Quantum mechanics
So far we have considered quantum eld theory with 0-dimensional spacetime (to make a joke, one
may say that the dimension of the space is 1). In this section, we will move clos
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MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
6. Matrix integrals and counting planar diagrams
6.1. The number of planar gluings. Let us return to the setting of 4. Thus, we have a potential
U (x) = x2 /2 j 0 gj xj /j (with gj being formal par
MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
19
4. Matrix integrals
Let hN be the space of Hermitian matrices of size N . The inner product on hN is given by (A, B ) =
Tr(AB ). In this section we will consider integrals of the form
2
ZN = N /2
e
MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
27
5. The Euler characteristic of the moduli space of curves
Matrix integrals (in particular, computation of the polynomial Pm (x) can be used to calculate the
orbifold Euler characteristic of the mod
MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
1. Generalities on quantum field theory
1.1. Classical mechanics. In classical mechanics, we study the motion of a particle. This motion is
described by a (vector) function of one variable, q = q (t),
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MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
2. The steepest descent and stationary phase formulas
Now, let us forget for a moment that the integrals (1,2,3) are innite dimensional and hence problematic to dene, and ask ourselves the following
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MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY
10. Quantum mechanics for fermions
10.1. Feynman calculus in the supercase. Wicks theorem allows us to extend Feynman calculus
to the supercase. Namely, let V = V0 V1 be a nite dimensional real sup