T ψθ (x) ≡ ψθ (x + X) = eiθ ψθ (x). (85)

We ¬rst restrict space to a box of size N X with periodic boundary conditions.

This implies a quantization of the angle θ

eiN θ = 1 ’ θ = θp ≡ 2πp/N , 0¤p<N.

We call ψp,n the normalized eigenfunctions of H corresponding to the band n

and the pseudo-momentum θp ,

NX

—

dx ψp,m (x)ψq,n (x) = δmn δpq ,

0

and En (θp ) the corresponding eigenvalues. Reality implies

En (θ) = En (’θ).

This leads to a decomposition of the identity operator in [0, N X]

—

δ(x ’ y) = ψp,n (x)ψp,n (y).

p,n

230 J. Zinn-Justin

We now consider an operator O that commutes with T :

[T, O] = 0 ’ x| O |y = x + X| O |y + X .

Then,

NX

—

q, n| O |p, m = dx dy ψq,n (x) x| O |y ψp,m (y) = δpq Omn (θp ).

0

Its trace can be written as

NX X

dx x| O |x = N dx x| O |x =

tr O = Onn (θp ).

0 0 p,n

We then take the in¬nite box limit N ’ ∞. Then,

2π

1 1

’ dθ

N 2π 0

p

and, thus, we ¬nd

X 2π

1

dx x| O |x = Onn (θ)dθ . (86)

2π

0 0

n

We now apply this general result to the operator

O = T e’βH .

Then,

X 2π

1

’βH

|x dx = ei θ’βEn (θ)

x| T e dθ ,

2π

0 0

n

which using the de¬nition of T can be rewritten as

X 2π

1

’βH

|x dx = ei θ’βEn (θ)

x + X| e dθ .

2π

0 0

n

In the path integral formulation, this leads to a representation of the form

2π

1

ei θ’βEn (θ)

[dx(t)] exp [’S(x)] = dθ ,

2π

x(β/2)=x(’β/2)+ X 0

n

where x(’β/2) varies only in [0, X], justifying the representation (43).

Chiral Anomalies and Topology 231

Appendix B.

Resolvent of the Hamiltonian in Supersymmetric QM

The resolvent G(z) = (H + z)’1 of the hermitian operator

H = ’d2 + V (x),

x

where ’z is outside the spectrum of H, satis¬es the di¬erential equation:

’d2 + V (x) + z G(z; x, y) = δ(x ’ y) . (87)

x

We recall how G(z; x, y) can be expressed in terms of two independent solutions

of the homogeneous equation

’d2 + V (x) + z •1,2 (x) = 0 . (88)

x

If one partially normalizes by choosing the value of the wronskian

W (•1 , •2 ) ≡ •1 (x)•2 (x) ’ •1 (x)•2 (x) = 1

and, moreover, imposes the boundary conditions

•1 (x) ’ 0 for x ’ ’∞, •2 (x) ’ 0 for x ’ +∞ ,

then one veri¬es that G(z; x, y) is given by

G(z; x, y) = •1 (y)•2 (x) θ(x ’ y) + •1 (x)•2 (y) θ(y ’ x) . (89)

After some algebra, one veri¬es that the diagonal elements G(z; x; x) satisfy a

third order linear di¬erential equation.

If the potential is an even function, V (’x) = V (x),

•2 (x) ∝ •1 (’x).

Application. We now apply this result to the operator

H = DD† with z = k 2 .

The functions •i then satisfy

DD† + k 2 •i (x) ≡ ’d2 + A2 (x) + A (x) + k 2 •i (x) = 0 ,

x

and (89) yields the resolvent G’ (k 2 ; x, y), related to the matrix elements (84)

by

G22 (k 2 ; x, y) = ’kG’ (k 2 ; x, y).

The corresponding solutions for the operator D† D + k 2 follow since

D† DD† + k 2 •i = 0 = D† D + k 2 D† •i = 0 .

232 J. Zinn-Justin

The wronskian of the two functions

χi (x) = D† •i (x),

needed for normalization purpose, is simply

W (χ1 , χ2 ) ≡ χ1 (x)χ2 (x) ’ χ1 (x)χ2 (x) = ’k 2 .

Thus, the corresponding resolvent G+ (in (84) G11 = ’kG+ ) reads

1

G+ (k 2 ; x, y) = ’ [χ1 (y)χ2 (x) θ(x ’ y) + χ1 (x)χ2 (y) θ(y ’ x)] .

k2

The limits x = y are

1

G+ (k 2 ; x, x) = ’

G’ (k 2 ; x, x) = •1 (x)•2 (x), χ1 (x)χ2 (x).

k2

If the potential is even, here this implies that A(x) is odd, G± (k 2 ; x, x) are even

functions.

We also need D† G’ (k 2 ; x, y):

D† G’ (k 2 ; x, y) = •1 (y)D† •2 (x)θ(x ’ y) + •2 (y)D† •1 (x)θ(y ’ x).

We note that D† G’ (k 2 ; x, y) is not continuous at x = y:

lim D† G’ (k 2 ; x, y) = •2 (x)D† •1 (x), lim D† G’ (k 2 ; x, y) = •1 (x)D† •2 (x)

y’x+ y’x’

and, therefore, from the wronskian,

lim D† G’ (k 2 ; x, y) ’ lim D† G’ (k 2 ; x, y) = 1 .

y’x’ y’x+

The half sum is given by

D† G’ (k 2 ; x, y) + D† G’ (k 2 ; x, y)

D† G’ (k 2 ; x, x) = 1 1

lim lim

2 y’x 2 y’x

’ +

= 1 D† •1 (x)•2 (x) + 1 D† •2 (x)•1 (x)

2 2

1

= (•1 •2 ) (x) + A(x)•1 (x)•2 (x).

2

This function is odd when A(x) is odd.

In the limit k ’ 0, one ¬nds

∞

x

du e’2S(u) , du e’2S(u)

•1 (x) = N eS(x) •2 (x) = N eS(x)

’∞ x

with

+∞

du e’2S(u) = 1 .

N2

’∞

Moreover,

D† •1 (x) = ’N e’S(x) , D† •2 (x) = N e’S(x) .

Chiral Anomalies and Topology 233

Therefore, as expected

1 2 ’S(x)’S(y)

G+ (k 2 ; x, y) ∼ Ne .

k’0 k 2

Finally,

y

D† G’ (0; x, y) = N 2 θ(x ’ y)e’S(x)+S(y) du e’2S(u) + (x ” y)

’∞

and, therefore,

∞

dt sgn(x ’ t)e’2S(t) .

D† G 12

’ (0; x, x) = 2N

’∞

References

1. The ¬rst part of these lectures is an expansion of several sections of J. Zinn-

Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press (Oxford

1989, fourth ed. 2002), to which the reader is referred for background in particular

about euclidean ¬eld theory and general gauge theories, and references. For an

early reference on Momentum cut-o¬ regularization see W. Pauli and F. Villars,

Rev. Mod. Phys. 21 (1949) 434.

2. Renormalizability of gauge theories has been proven using momentum regulariza-

tion in B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3121, 3137, 3155; D7

(1973) 1049.

3. The proof has been generalized using BRS symmetry and the master equation

in J. Zinn-Justin in Trends in Elementary Particle Physics, ed. by H. Rollnik

and K. Dietz, Lect. Notes Phys. 37 (Springer-Verlag, Berlin heidelberg 1975); in

Proc. of the 12th School of Theoretical Physics, Karpacz 1975, Acta Universitatis

Wratislaviensis 368.

4. A short summary can be found in J. Zinn-Justin Mod. Phys. Lett. A19 (1999) 1227.

5. Dimensional regularization has been introduced by: J. Ashmore, Lett. Nuovo Ci-

mento 4 (1972) 289; G. ™t Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189;

C.G. Bollini and J.J. Giambiagi, Phys. Lett. 40B (1972) 566, Nuovo Cimento 12B

(1972) 20.

6. See also E.R. Speer, J. Math. Phys. 15 (1974) 1; M.C. Berg`re and F. David, J.

e

Math. Phys. 20 (1979 1244.

7. Its use in problems with chiral anomalies has been proposed in D.A. Akyeampong

and R. Delbourgo, Nuovo Cimento 17A (1973) 578.

8. For an early review see G. Leibbrandt, Rev. Mod. Phys. 47 (1975) 849.

9. For dimensional regularization and other schemes, see also E.R. Speer in Renormal-

ization Theory, Erice 1975, G. Velo and A.S. Wightman eds. (D. Reidel, Dordrecht,

Holland 1976).