Supersymmetric Quantum Field Theory

in 1-D

Saba Asif Baig

A Thesis Submitted in partial fulfillment for the

Degree of Bachelors of Science in Physics at Lums

Adviser: Dr. Babar Qureshi

May 2017

c� Copyright by Saba Asif Baig, 2017.

All rights reserved.

Abstract

This thesis is an introductory study of supersymmetric quantum field theory in 1-

dimension. In particular, the thesis is focused on exploring the supersymmetric al-

gebra for generalized quantum systems. This is used to deduce two key features of

the system: the mapping between the bosonic and fermionic sub-spaces of a system

and the nature of the ground state of the model. Specifically, the ground state of

the system is analysed by making use of the Witten index, and this is computed in

both operator and path integral formalism. The latter of the two makes use of the

localization principle, and the thesis explores this in 0 dimension to exhibit it as a

feature of any general higher dimension supersymmetric theory. This is then applied

in the 1 dimensional case to show results identical to those produced in operator

formalism. The aim of this thesis is to show the importance of the localization prin-

ciple in solving path integrals. The result of the thesis finds a useful application in

supersymmetric quantum field theory in 1+1 dimension where the integrals are not

reducible otherwise.

iii

Acknowledgements

I would like to thank my advisor, Dr. Babar Qureshi for his continued support,

motivation and encouragement of my interests in theoretical high energy physics,

throughout the course of my Undergaduate program. I would also like to thank Dr.

Adam Zaman for the numerous courses he has taught: I learned a lot in the courses

and acquired a taste for rigorous physics from there. A special mention to my first

reviewer and critic of all my works, my brother Mohammad Haris Baig, for always

having faith in me and my thesis, and also, to my fellow physics seniors who made

the experience memorable.

iv

To my parents

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Classical Supersymmetric Action 1

1.1 Action for Bosonic Harmonic Oscillator . . . . . . . . . . . . . . . . . 1

1.2 Action for Fermionic Harmonic Oscillator . . . . . . . . . . . . . . . . 3

1.3 Generalized Supersymmetric Action . . . . . . . . . . . . . . . . . . . 6

2 Quantization of the Supersymmetric System 9

2.1 Supersymmetric algebra . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Witten Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Witten Index using Partition Function in Path Integral Formalism 18

3.1 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Bosonic coherent states . . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Fermionic coherent states . . . . . . . . . . . . . . . . . . . . 21

3.2 Fermionic path integrals . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Localization Principle 29

4.1 Localization Principle in 0-D, using Supersymmetric Transformations 29

4.2 Path integrals in 1-dimension . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Localization Principle in 1-D . . . . . . . . . . . . . . . . . . . . . . . 37

vi

5 Conclusion 40

Bibliography 42

vii

Chapter 1

Classical Supersymmetric Action

The action integral is one of the most useful tools of classical mechanics, and the

purpose of this chapter is to motivate a general classical action that is invariant under

supersymmetric transformations using the simple and familiar case of a harmonic

oscillator.

1.1 Action for Bosonic Harmonic Oscillator

The bosonic harmonic oscillator is well understood in both, classical physics and

quantum mechanics, and can be solved to give exact solutions, making it a good

starting point since we can verify our results.

The hamiltonian for a bosonic harmonic oscillator in phase space (co-ordinate and

conjugate momentum) is given by:

Hb =1

2p2 +

1

2w2x2 (1.1)

1

This hamiltonian can be expressed in variables, a and its complex conjugate a, which

are complex combinations of the co-ordinate and momentum variables.

a =1p2w

(wx + ip) a =1p2w

(wx - ip)

Hb = waa (1.2)

The action for this hamiltonian, using the variable a and a, is derived using reverse

Legendre transformation.

S =

ZL dt where L = pq �H and p =

@L

@q(1.3)

The conjugate momentum for the variable a is found using the Poisson brackets, as

defined in classical mechanics:

{A,B}PB =Xi

(@A

@qi

@B

@pi� @A

@pi

@B

@qi) where {qk, pj}PB = �kj

pa = ia

As a and a are complex conjugates of one another, it will be su�cient to find the

conjugate momentum for any one variable and use that to reverse Legendre transform

to an action.

The action, up to boundary terms, can be expressed as:

S[x, p] =

Zdt (px� 1

2(p2 � w2x2))

S[a, a] =

Zdt (iaa� waa) (1.4)

2

1.2 Action for Fermionic Harmonic Oscillator

The dynamics of the fermionic harmonic oscillator can be described using a similar

action but making the use of complex Grassmann valued functions of time, as the

analogs of a and a. These will better represent fermionic variables (and later fermionic

operators upon quantization) even at the classical level, under Pauli’s exclusion prin-

ciple according to which no two identical fermions can be put in the same state. The

Grassmann algebra is given below in substantial detail since it is extensively used

ahead. [3]

{✓i, ✓j} = 0 implying ✓2i = 0 (1.5)

The implication means that the only non-zero terms involving a fermionic variable

are those where the variable has a power of 0 or 1, but no more: this is the essence of

Pauli’s exclusion principle. Functions of fermionic variables can always be expressed

as a sum of the di↵erent fermionic terms. For the case of two variables, this can be

expressed as:

f(✓1, ✓2) = f0 + f1✓1 + f2✓2 + f3✓1✓2

Terms with an even number of variables are called Grassmann even, and commute

with a single fermionic variable, as the even number of negative signs resulting from

each switch cancel. Terms with an odd number of variables are called Grassmann

odd, and anti-commute with a single fermionic variable.

Derivatives with respect to a Grassmann variable are easy as a function can at most

be linear in it’s dependence on a variable. Left derivatives are defined as removing

the variable from the left, and right derivatives are defined as removing the variable

from the right. The left and right derivatives will be specified as used in the course

3

of the thesis.

@Lf(✓1, ✓2)

@✓1= f1 + f3✓2

@Rf(✓1, ✓2)

@✓1= f1 � f3✓2 (1.6)

Grassmann integration is defined by Berezin, as identicial to di↵erentiation:

Zd✓f(✓) =

@Lf(✓)

@✓(1.7)Z

d✓1f(✓1, ✓2) =@L(f0 + f1✓1 + f2✓2 + f3✓1✓2)

@✓1= f1 + f3✓2

The complex conjugate for two real Grassmann variables is purely imaginary as shown

below:

✓ = ✓ so ✓1✓2 = ✓2✓1 = ✓2✓1 = �✓1✓2 (1.8)

Since the action integral for the bosonic cases refers to variables as functions of time,

the fermionic analog will contain ✓(t) and ✓(t0), which will be referred to as di↵erent

fermionic variables, like ✓1 and ✓2. This implies:

✓(t)✓(t) = 0 and ✓(t0)✓(t) = �✓(t)✓(t0) (1.9)

Time derivatives of Grassmann variables and their algebra is as follows:

˙✓(t) =✓(t0)� ✓(t)

t0 � t

˙✓(t) ˙✓(t) =✓(t0)✓(t0)� ✓(t0)✓(t)� ✓(t)✓(t0) + ✓(t)✓(t)

(t0 � t)2= 0

˙✓(t)✓(t) =✓(t0)✓(t)

t0 � t=

�✓(t)✓(t0)t0 � t

= �✓(t) ˙✓(t) (1.10)

4

The fermionic action’s form is assumed to be similar to it’s bosonic counterpart,

using complex grassman variables (t) and ¯ (t) instead of a and a:

S[ , ] =

Zdt (i � w ) (1.11)

The action is real upto boundary terms, and is easily verified using integration by

parts.

Note, that the ordering of the reverse Legendre transform as given in Equation

1.3, is taken such that the conjugate momentum for Grassmann variables, denoted

by ⇡, is obtained correctly from the Lagrangian using the left derivative convention.

This conjugate momentum should satisfy the classical Poisson bracket relationship:

{⇡ , }PB = (@⇡ @

@

@⇡ � @⇡

@

@⇡ @

) = �1 (1.12)

⇡ =@LL

@ = �i (1.13)

{�i , }PB = i{ , }PB = �1 implying { , }PB = �i (1.14)

Since the conjugate momentum ⇡ turns out to be a multiple of , and in the La-

grangian is multiplying by , the ordering has to be taken carefully during Legendre

transformation in sync with the defined derivative, or we would have an incorrect

sign, based on the fact that and it’s derivative do not commute. Also, the poisson

bracket is the first place that provides a relationship between and . We did not

impose any condition on it prior to this.

To compute the Hamiltonian for this case, we use the Legendre transformation,

and the anti-commuting property of a Grassmann variable and its derivative.

H = ⇡ � L (1.15)

H = �i �i + w = w (1.16)

5

1.3 Generalized Supersymmetric Action

The bosonic and fermionic harmonic oscillator hamiltonians from the last two sections

are added to give the supersymmetric harmonic oscillator’s hamiltonian:

Hs = Hb +Hf =1

2p2 +

1

2w2x2 + w (1.17)

This hamiltonian is expressed in a generalized form using a pre-potential W(x): [6]

Hs =1

2p2 +

1

2(W 0(x))2 +W 00(x) (1.18)

where W (x) =1

2wx2 (1.19)

This motivates us to consider this as the general form of any supersymmetric hamil-

tonian. However, we will write it in a more symmetric form, where the fermionic

variables and occurr in both orderings, by making use of their anti-commuting

property.

Hs =1

2p2 +

1

2(W 0(x))2 +

1

2W 00(x)( � ) (1.20)

Legendre transformation of the Hamiltonian in Equation 1.18, and using the conju-

gate momentum as defined in Equation 1.13 gives us the supersymmetric classical

Lagrangian:

L = xp+ ⇡ �H

L =1

2(x)2�1

2(W 0(x))2 + i �W 00(x) (1.21)

where x(t) is the co-ordinate of the particle in one-dimension, and & are the

Grassmann variables describing the additional degrees of freedom.

6

The lagrangian’s i term can be expressed in more symmetric terms, by making

use of the property in Equation 1.8. Hence, the classical supersymmetric lagrangian

is given by:

L =1

2(x)2 � 1

2(W 0(x))2 +

1

2i( � ˙ )�W 00(x) (1.22)

The action integral is defined below, along with invariance of the action integral, in

the usual terms of variational calculus, except that care has to be taken for fermionic

variables with respect to their derivatives, integrals and anti-commutation relations.

S[x, , ] =

ZL dt , �S =

Z�L dt = 0 (1.23)

The action integral is invariant under the following transformations, resulting in con-

served charges, as per Noether’s theorem: [3]

• Time translation: t0 = t-a gives Hamiltonian H

�x = ax

� = a

� = a ˙

• U(1) phase transformations, with parameter ↵ gives Fermion number F

�x = 0

� = i↵

� = �i↵

7

• Infinitesimal supersymmetric transformations with fermionic parameter ✏ and ✏

gives supercharges Q and Q

�x = i(✏ + ✏ )

� = �✏(x� iW 0(x))

� = �✏(x+ iW 0(x))

The corresponding conserved charges for the above transformations are:

H = 12p

2 + 12(W

0(x))2 +W 00(x)

F =

Q = (x+ iW 0(x))

Q = (x� iW 0(x))

The characterizing property of supersymmetry is that the composition of two

supersymmetries generates a time translation, as exhibited in the Poisson bracket

connecting the conserved supercharges Q and Q with the Hamiltonian. This is done

by substituting x with p, to obtain the charges in phase space. The commutation

relations in the phase space, and the Poisson bracket algebra are given:

{x, p}PB = 1 { , } = �i

{Q, Q}PB =� 2iH (1.24)

{F,Q}PB = iQ (1.25)

{x, Q}PB =� iQ (1.26)

8

Chapter 2

Quantization of the

Supersymmetric System

In section 1.3, we considered the hamiltonian and the lagrangian in their symmetric

forms, that is their fermionic variables were present in both orderings. Addition-

ally, for the bosonic case, the lagrangian can be expressed in sympletic terms (where

the time derivative is equally shared between the co-ordinate and the conjugate mo-

mentum). In this particular form, the lagrangian expressions for the fermionic and

bosonic cases motivates a graded algebra. This graded algebra is in keeping with the

properties of the Poisson brackets, and makes it consistent to adopt the following

canonical quantization rule, that will promote phase space co-ordinates to operators

(denoted by hats), with their commutation/anti-commutation relations fixed by their

classical Poisson brackets: [3]

[ZA, ZB} = ih{ZA, ZB}PB (2.1)

[·, ·} =

8>><>>:{·, ·} anti-commutator if both variables are fermionic

[·, ·] commutator otherwise

9

For simplicity’s sake, we usually set h = 1.For the bosonic variables, x and p:

{x, p}PB = 1 �! [x, p] = i (2.2)

For the fermionic variables, & :

{ , }PB = �i �! { , †} = 1 (2.3)

This quantization results in the following fermionic anti-commutator rules, in terms

of now operators and †:

{ , †} = 1 (2.4)

{ , } = 0 { †, †} = 0 (2.5)

Then, to write the quantized Hamiltonian, we have to promote the phase space

co-ordinates in Equation 1.20 to operators. The reason for writing Equation 1.20 such

that consists of both the orderings of and , was to ensure that the variables be eas-

ily promoted to operators, without any problems of any factors. The supersymmetric

quantized hamiltonian is given by:

Hs =1

2p2 +

1

2(W 0(x)2 +

1

2W 00(x)( † � †) (2.6)

The operators have hats (ˆ) and the daggers (†) denotes complex conjugate operators.

2.1 Supersymmetric algebra

This generalized supersymmetric quantized system will have the same conserved

charges as it’s classical analog in section 1.3. The di↵erence is that they are gen-

erate by their corresponding operators and therefore, the conserved quantities are

10

themselves operators. We can again simply promote the co-ordinates to operators

since we have obtained all of them by using invariance of the lagrangian that has

both the fermionic orderings within it.

These conserved charges are:

F = †

Q = ( ˙x+ iW 0(x))

Q† = ( ˙x� iW 0(x) †

Supersymmetric algebra is the algebra that we will derive using the commutators and

anti-commutators linking the various conserved charges.

The operator F is defined as the fermionic number operator, and has the following

commutation relation with the fermionic variables:

[F , ] = � , [F , †] = † (2.7)

Here, and † can be considered as the fermionic lowering and raising operators.

Using the principle of quantized states as applied in the harmonic oscillator, we define

a zero state, as the satte annihilated by the lowering operator: |0i = 0 Then, the

rest of the states are built by acting the raising operator onto |0i. However, since

( †)2 = 0, the only other possible state that can be constructed is: † |0i. The

fermionic Hilbert is therefore only two-dimensional.

The total Hilbert space of states and the fermionic operators can be expressed in

this two dimensional basis as: [1]

H = L2(R)O

F2 = L2(R) |0i+ L2(R) |1i = HB +HF

|0i = 0 h0, 0i = 1 (Fock vacuum) |1i = † |0i (2.8)

11

=

0B@ 0 1

0 0

1CA † =

0B@ 0 0

1 0

1CA (2.9)

The total Hilbert space can be divided into two subspace, a bosonic subspace and

a fermionic subspace. The operator F acts on HB to give zero and on HF to given

one. All three of the above conserved charges, F , Q, and Q† commute with the

hamiltonian. In the case of the supercharges, this implies that energy of the state

is not modified by an action of the supercharge; their action spans a degeneracy in

energy.

[Q, H] = [Q†, H] = [F , H] = 0 (2.10)

Q and Q† are the two conserved supercharges that can generate the fermionic

symmetry transformations shown in section 1.3 :

For O = O(x, , ) we have:

�O = [�,O], � = ✏Q + ✏ Q†

These conserved supercharges satisfy the following commutation relations with the

Fermionic number operator:

[F , Q] = Q , [F , Q†] = �Q† (2.11)

Q(�1)F =� (�1)F Q , Q†(�1)F = �(�1)F Q†

From the above equation, it is easily evident that the supercharges map the bosonic

states to the fermionic states and vice versa, since their action causes a change in

the fermion number. There is mapping between the bosonic and fermionic hilbert

spaces using these conserved supercharges. Equations 2.10 and 2.11 can cumulatively

be used to show that for a pair of degenerate energy states, one will belong to the

bosonic hilbert space and one to the fermionic hilbert space. Hence, supercharges are

12

the generators of supersymmetric transformations since they allow the states to move

in between the two hilbert spaces.

Commutation relations are give by:

{Q, Q†} = 2H (2.12)

{Q, Q} = 0 , {Q†, Q†} = 0 (2.13)

A characteristic feature of supersymmetry is that the hamiltonian generates time

translations, which are obtained by two supersymmetry transformations.

The rest of the graded commutation relations are all vanishing.

We can express the an eigenfunction of the state, and all four of the conserved

charges in the above basis. This is given below: [2]

(x) =

0B@ �1(x)

�2(x)

1CA (1)

x = x p = �i @@x

Q =

0B@ 0 �i@x + iW 0(x)

0 0

1CA (2)

Q† =

0B@ 0 0

�i@x � iW 0(x) 0

1CA (3)

F =

0B@ 0 0

0 1

1CA (4)

H =

0B@ �12@x

2 + 12(W

0(x))2 � 12W

00(x) 0

0 �12@x

2 + 12(W

0(x))2 + 12W

00(x)

1CA (5)

13

Generally, any supersymmetric theory will satisfy the following algebriac struc-

ture:

{Q,Q } = 2H, {(�1)F ,Q } = 0, [(�1)F ,H] = 0

For the above supersymmetric theory with the given conserved supercharges, we use

our Q1 as defined in Q = Q1 + iQ2 to satisfy the requirements.

The following properties of the Hilbert space can be easily deduced using the same

analaysis as employed in regular quantum mechanics, of commutators of di↵erent

operators with the hamiltonian H, and amongst themselves:

• The Hamiltonian H is positive definite.

• Any state | 0i with energy E=0, is neccesssarily a ground state and super-

symmetric, that is, it is annihilated by both operators Q and Q. Hence, a

supersymmetric state is one that cannot be mapped from the bosonic hilbert

subspace to the fermionic one or vice versa: it is invariant under the action of

these supercharges.

• Energy levels with E 6= 0, are degenerate, and split evenly within the bosonic

and fermionic sub-spaces.

This is used to deduce that all energy levels with E 6= 0 are degenerate, with an equal

number of bosonic and fermionic states, while states with E = 0 are supersymmetric

(singlets under supersymmetry) and, if they exist, they are possible ground states of

the model.

14

2.2 Witten Index

Witten index: A useful definition is: [4]

n(E=0)b � n

(E=0)f = Tr[(�1)F exp��H ] = TrE=0[(�1)F ]

1. This can be easily seen, by working out the trace over the entire energy eigen-

states in the hilbert space. For every E 6= 0, there is a degenerate pair of states:

one bosonic and one fermionic which results in the same value of the exponent,

while the corresponding (�1)F results in a plus and a minus sign respectively,

causing the term to cancel to 0, for all n 6= 0.

2. A single state cannot leave the E = 0 energy level as it must have a partner to

form a doublet degenerate in energy for any of the higher n levels.

3. Only pairs of states can leave the zero energy level by acquiring a small value

of the energy, so that they can form a supersymmetric doublet.

4. If paired states with E 6= 0, by varying the parameters, acquire the exact value

E = 0, they do not modify the value of the Witten index, since one is bosonic

and one is fermionic. Hence, the Witten index is invariant under any of these

deformations. The idea behind it is that the nature of the ground state of a

system is usually preserved in any evolution of the above kind.

To Compute the Witten Index and it’s cases for N = 2 model, the vacuum states

| 0i with E = 0 must satisfy the following condition:

H | 0i = 0

15

Using the anti-commutator connecting the supercharges operators to the Hamiltonian,

from Equation 2.12, this condition can be re-expressed as:

Q| 0 >= 0 and Q†| 0 >= 0 (2.14)

Solving this using the representation given above from 1 to 5, results in:

0(x) =

0B@ c1e�W (x)

c2eW (x)

1CAThe normalizability condition can be used to discuss the solution by looking at three

cases:

• Case 1: W(x) ! 1 as x ! ±1

0(x) =

0B@ c1e�W (x)

0

1CAF 0(x) = 0

Witten index = 1

• Case 2: W(x) ! �1 as x ! ±1

0(x) =

0B@ 0

c2e�W (x)

1CAF 0(x) = 0(x)

Witten index = -1

Case 3: W(x) ! ±1 or ⌥1 as x ! ±1For this case, we have no normalizable solutions, and the Witten index vanishes.

16

If there are no states with E = 0, then supersymmetry is spontaneously broken:

the ground state, whichever one it may be, is a state that is not invariant under

supersymmetric transformations. This happens in the third case above. If the Witten

index is non-zero, supersymmetry cannot be spontaneously broken: there always

exists at least one ground state with E = 0, which is necessarily invariant under

supersymmetry. Based on the Witten index, +1 or �1, it is bosonic or fermionic in

nature but cannot be mapped by the conserved supercharge operators between the

bosonic and fermionic hilbert sub-spaces.

17

Chapter 3

Witten Index using Partition

Function in Path Integral

Formalism

The purpose of this chapter is to show that the Witten index which is given by

the partition function in terms of the transition amplitudes is equivalent to its path

integral formalism, for both bosonic and fermionic variables. To show this for the

fermionic case, a reference to the concept of coherent states is needed as this will the

derivation of the path integral form easier and will be the first showing of the Witten

index. [3]

3.1 Coherent states

In the fermionic Fock basis, coherent states are eigenstates of the fermionic operator

, with Grassmann valued eigenvalues, and form an overcomplete basis. These will

then be used to convert the transition amplitude’s matrix elements into a path integral

over some Grassmann functions. To find these for the fermionic case, lets first look

at the construction of the bosonic coherent states.

18

3.1.1 Bosonic coherent states

In the bosonic case, the creation and annihilation operators are given by a† and a,

where they follow they follow the commutation relations, as derived in the previous

chapters, and are respective used to raise and lower the states.

For the fock case, a complete orthonormal basis can be constructed by defining

the Fock vacuum, and then all other states in terms of the Fock vacuum. This is done

as follows:

|0i where |0i = 0 (3.1)

|1i = a† |0i (3.2)

|2i = a†p2|1i = (a†)2p

2!|0i (3.3)

|ni = a†pn|n� 1i = (a†)np

n!|0i (3.4)

First, please note that this is not the coherent set of states, since these states are

evidently not eigenstates of the creation or annihilation operators. Furthermore, for

normalization, we normalize the Fock vacuum, so that h0| |0i = 1, and use this in

conjunction with the definition of the basis states above to show that these states are

orthonormal:

hm| |ni = h0| (a)n

pm!

(a†)npn!

|0i = �n,m (3.5)

In the above, n and m obviously take integer values. While this may be a long result

to show for large m and n, it can be easily shown for small m and n that take values

between 0 and 2.

Now to make coherent states, denoted by |ai, the requirement is for them to be

eigenstates of the annihilation operator, with the eigenvalue in general being complex,

denote by a. Hence, we come up with a suitable construction of the coherent state,

17

and show below that it indeed is an eigenstate of the annhilation operator.

a |ai = a |ai where |ai = eaa† |0i (3.2)

a† |ni = pn+ 1 |n+ 1i and a |ni = p

n |n� 1i (3.3)

a |ai = a(eaa† |0i) (3.4)

= a[(1 + aa† +(aa†)2p

2!+ ...+

(aa†)npn!

+ ...) |0i] (3.5)

= a[|0i+ a |1i+ a2p2!

|2i+ ...+anpn!

|ni+ ...] (3.6)

= 0 + a |0i+ a2 |1i+ ...+anp

(n� 1)!|n� 1i+ ... (3.7)

= a[0 + |0i+ a |1i+ ...+an�1p(n� 1)!

|n� 1i+ ...] (3.8)

= a(eaa† |0i) (3.9)

There are now a set of properties that can be easily shown for these coherent states,

using regular quantum mechanical methods. They follow quite simply through our

usual understanding of operator formalism.

ha| = |ai† ha| a† = ha| a (3.10)

ha| |ai = eaa Scalar product (3.11)

hb| |ai = eba 6= 0 (3.12)

1 =

Zdada

2⇡ie�aa |ai ha| Identity (3.13)

Tr A =

Zdada

2⇡ie�aa ha| A |ai Trace of operator A (3.14)

These coherent bosonic states form an over-complete basis, as is eveident by the

third property shown above, which shows that two states hb| |ai are not orthonormal.

20

3.1.2 Fermionic coherent states

In a similar manner, we can develop a set of coherent states for the fermionic system.

The fermionic anti-commutation relations of the femrionic creation and annhilation

operators are the same as stated before. The original fock basis for the fermionic is

already developed in the previous chapter and consists of two possible kets.

Similar to the bosonic case, we will define | i as the eigenstate of the annihilationoperator, with a general complex Grassman number as an eigenvalue. This is given

below:

| i = | i (3.15)

Now, note that the grassman numbers and its complex conjugate anti-commute

with oneanother. However, upon quantization, the annhilation and creation opera-

tors, and †,follow a given anti-commutator - derived in chapter 1. Now, we will

define that the Grassmann numbers will anti-commute with the fermionic operators.

Some properties of these coherent states can be developed and will be useful ahead

in determining the fermionic partition function in path integral formalsim.

The most important is to define the coherent state itself. A definition is proposed

below and then proven to be the eigenstate of the annihilation operator. Care must be

taken that the grassmann eigenvalue and the fermionic operators cannot be exchanged

without taking note of the addition sign since they anti-commute.

| i = e † |0i (3.16)

| i = (1 + † ) |0i = |0i � † |0i = |0i � |1i (3.17)

| i = (|0i � |1i) = |1i = |0i (3.18)

= (|0i � |1i) using = 0 (3.19)

= | i (3.20)

21

This admission of the 2 term can be used, eliminated or added at convenience

to simplify and achieve di↵erent forms.

To define the bra operator, we take the complex conjugate of the coherent state.

This is done below, but taking care that complex conjugation reverses the order of

the grassman variables and/or operators.

| i† = (e † |0i)† = h0| e (3.21)

Furthermore, complex conjugation of the action of on the coherent state will be

more revealing:

( | i)† = | i† † (3.22)

| i† † = h0| e † = h0| (1 + ) † (3.23)

= 0 + h0| † = h1| † = h0| (3.24)

= (h0|� h1| ) = | i† (3.25)

From this, it can be seen that | i† is an eigenstate of the creation operator †, with

eigenvalue . Hence it makes sense to denote | i† as shown below:

| i† = h | = h0| e † (3.26)

h | † = h | (3.27)

22

A scalar product can be defined as follows:

h | | i = (h0|� h1| )(|0i � |1i) (3.28)

= h0| |0i � h1| |0i � h0| |1i+ h1| |1i (3.29)

= 1� h1| |0i � h0| |1i+ h0| † |0i (3.30)

= 1� 0� 0 + h0| † |0i = (3.31)

1 + h1| |1i = 1 + = e (3.32)

The identity is a very useful quantity that is often entered into expressions to

simplify. We will once again suggest an expression, and show that it equated that

of a generalized notion of identity. It will be useful to remember that integration

over a grassmann variable is defined by equating it to the di↵erentiation by the same

grassmann variable (left derivative as taken by convention).

Zd d e� | i h | = 1 =

Zd d (1� )(|0i � |1i)(h0|� h1| ) (3.33)

= |0i h0|+ |1i h1| (3.34)

As you will see, some of the most useful properties can be found by computing

the trace of some operator or another. For this purpose, we will suggest certain

expressions and prove their equivalence to those given for trace and supertrace. Some

common techniques of simplification used include the square of either of the fermionic

operators is equal to 0, and integration over d d will require both and to present

in any non-zero term. The following properties will be computed for bosonic operator

23

A.

Zd d e� h� | A | i

=

Zd d (1� )(h0|+ h1| )A(|0i � |1i)

=

Zd d (1� )(h0| A |0i � h0| A |1i+ h1| A |0i � h1| A |1i)

=

Zd d (1� )(h0| A |0i � h0| A |1i � h1| A |0i � h1| A |1i)

=

Zd d (� h1| A |1i � h0| A |0i)

=

Zd d ( h1| A |1i+ h0| A |0i)

= h0| A |0i+ h1| A |1i = Tr A

The supertrace is given by the following expression, where the operator F is the

fermionic number operator previously introduced that has an eigenvalue of 0 for |0iand an eigenvalue of 1 for |1i.

Str A = Tr [(�1)F A] (3.35)

= h0| (�1)F A |0i+ h1| (�1)F A |1i (3.36)

= h0| A |0i � h1| A |1i (3.37)

24

Then analogous to what’s done for the trace, an expression is prescribed below and

equivalence to the supertrace of it is shown.

Zd d e� h | A | i

=

Zd d (1� )(h0|� h1| )A(|0i � |1i)

=

Zd d (1� )(h0| A |0i � h0| A |1i � h1| A |0i+ h1| A |1i)

=

Zd d ( h1| A |1i � h0| A |0i)

= h0| A |0i � h1| A |1i = Str A

3.2 Fermionic path integrals

A partition function is given by the transition amplitudes in general between an initial

and a final state. Here we will use the transition amplitude between two coherent

states, given by h f | e�iHT | ii. The hamiltonian being considered is the fermionic

hamiltonian, expressed such that all creation operators are on the left side of the

annihilation operators. A hamiltonian can always be put in such a form given the

anti-commutation relations expressed earlier for fermionic operators. This will prove

to be useful since the coherent state | i is an eigenstate of the operator .

Transition amplitude is modified to it’s path integral formalism by making use of

small time steps and the identity decomposition in the coherent states basis. We will

25

split the time T into N steps, where each time step is ✏, such that: T = N✏.

h f | e�iHT | ii = h f | (e�iH✏)N | ii

= h f | e�iH✏e�iH✏.....e�iH✏ | ii

= h f | e�iH✏1e�iH✏1.....1e�iH✏ | ii

= h f | e�iH✏

Zd 1d 1e

� 1 1 | 1i h 1| e�iH✏

Zd 2d 2e

� 2 2 | 2i h 2| ....

....

Zd N�1d N�1e

� N�1 N�1 | N�1i h N�1| e�iH✏ | ii

=

Z(N�1Yk=1

d kd ke� k k)

NYk=1

h k| e�iH✏ | k�1i

To obtain the last step, we make use of the fact that the Hamiltonian always con-

tains grassmann even terms. This means that any of grassman numbers or di↵erential

terms from the identity expression can commute across it. Also, since the di↵erential

terms are in pairs in one identity, as are the grassman numbers, in crossing the kets

and bras that exist in the middle, no overall negative sign is accumulated. This is

how we clump the terms in the second last equality in the last form given and can

validly pull all the terms out as products. For the transition amplitude product, note

that 0 = i and N = f .

We can expand and simplify the transition amplitude given above. In order to

do this, it is important that the ordering of the hamiltonian is such that is on the

right and † is on the left so that they can act on their respective eigenvectors, | iand h |, and be replaced by the respective eigenvalues of .

h k| e�iH( †, )✏ | k�1i = h k| (1� iH( †, )✏+ ...) | k�1i (3.38)

= h k| | k�1i � i✏ h k| H( †, ) | k�1i+ ... (3.39)

= (1� i✏H( k, k�1) + ...) h k| | k�1i (3.40)

= e�i✏H( k, k�1) h k| | k�1i = e�i✏H( k, k�1)e k k�1 (3.41)

26

Plugging this back into the transition amplitude expression derived above this:

h f | e�iHT | ii =Z(N�1Yk=1

d kd ke� k k)

NYk=1

e�i✏H( k, k�1)e k k�1 (3.42)

=

Z(N�1Yk=1

d kd k)NYk=1

e� k k+ N N+ k k�1�i✏H( k, k�1) (3.43)

=

Z(N�1Yk=1

d kd k)eiPk=N

k=1 [i k k�i k k�1�H( k, k�1)✏] + N N (3.44)

=

Z(N�1Yk=1

d kd k)eiPk=N

k=1 [i k( k� k�1)

✏ �H( k, k�1)]✏ + N N (3.45)

=

Z(N�1Yk=1

d kd k)eiPk=N

k=1 [i k k�H( k, k�1)]✏ + N N (3.46)

In the limit that N approaches infinity, these discretized sums and products will be

replaced by integrals to give this final result:

h f | e�iHT | ii =Z

D D eiR T0 dt[i (t) (t)�H( (t), (t))] + (T ) (T ) (3.47)

=

ZD D eiS[ , ] (3.48)

where S =

Z T

0

dt[i (t) (t)�H( (t), (t))] � i (T ) (T ) (3.49)

Note that this S can be discretized and expressed in terms of k = (t) where t = k✏.

Also, the action in S has a boundary term, where (T ) = f and is important in

variational methods that vary the action integral by keeping the end points fixed,

that is i and f dont vary. In that sense, 3.52 basically represent a summation over

all paths of (t) and (t) with the endpoints fixed, and is weighed with the factor of

exponent of the action times i.

The partition integral in it’s path integral formalism is extremely useful. Here, we

will use this to derive the path integral form of the trace for the transition amplitude

e�iHt. The trace computed for a bosonic operator in fermionic variables contains a

27

transition amplitude part that can be substituted by its equivalent path integral form.

We shall compute this below for the trace in this case and follow it up by computing

it for the supertrace to show an interesting feature.

Tr[e�iHt] =

Zd 0d 0e

� 0 0 h� 0| e�iHt | 0i (3.50)

= limN!1

Z(N�1Yk=0

d kd k)eiPN

k=1[i k( k� k�1)

✏ �H( k, k�1)]✏ (3.51)

=

ZA

D D eiS[ , ] (3.52)

In the above expression we can see that the initial state is the negative of the

final state, that is the boundary condition is anti-periodic. In the continuum limit,

the trace is equivalent to the path integral sum over all anti-periodic paths: (T ) =

� (0). However, when checking for invariant actions under di↵erent symmetry trans-

formations in our 1 dimensional case of a circle of length �, the path begins and ends

at the same point so periodic boundary conditions are required. For this purpose, the

supertrace proves useful.

For the supertrace, the expression is given by:

Str [e�iHT ] =

Zd 0d 0e

0 0 h 0| e�iHT | 0i (3.53)

= limN!1

Z(N�1Yk=0

d kd k)eiPN

k=1[i k( k� k�1)

✏ �H( k, k�1)]✏ (3.54)

=

ZP

D D eiS[ , ] (3.55)

Above, the initial state is the same as final state, that is the boundary condition

is periodic. In the continuum limit, the trace is equivalent to the path integral sum

over all anti-periodic paths: (T ) = (0)

28

Chapter 4

Localization Principle

The aim of this chapter is to apply the localization principle to compute the Witten

index, given by the trace of (�1)F e�iHt, now using a path integral formalism. As

covered in the last chapter, we have already computed the path integral form for

the fermionic case, which is equivalent to the action integral over periodic boundary

conditions. The bosonic analog is the same expression (with periodic conditions).

However, the complete integral, over both bosonic and fermionic variables, is com-

plicated to solve in general in higher dimension. For this purpose, we resort to a

localization principle which uses super symmetric transformations to reduce the con-

tributions from all periodic paths to a smaller dimension. In the case of 0 and 1

dimension, we will see the reduced integral only has contributions from certain fixed

points of the fermionic supersymmetric transformations.

4.1 Localization Principle in 0-D, using Supersym-

metric Transformations

The action integral in general has to have an even number of fermionic variables in

any of its terms (Grassman even), and using the Grassmann algebra and integration

29

as discussed in the above section, the smallest number of fermionic variables for a

non-trivial Hamiltonian is 2. A general action in this case can expressed as: [4]

S(X, 1, 2) = S0(X)� 1 2S1(X) (4.1)

The partition function is given by:

Z =

Z Yi

dX iYa

d a e�S(X, )

=

ZdXd 1d 2e�S0(X)+ 1 2S1(X)

=

ZdXd 1d 2e�S0(X)(1 + 1 2S1(X))

=

ZdXd 1d 2e�S0(X) +

ZdXd 1d 2e�S0(X) 1 2S1(X))

= 0 +

ZdXe�S0(X)S1(X)

The partition function therefore allows you to integrate out any fermionic variables

leaving behind the path integral over bosonic variables.

We will proceed with a special action now, consider a supersymmetric transforma-

tion for it and use that to show the reduction of the path itegral in general. For this

purpose, we will not use the only bosonic reduction of the path integral, since keeping

the fermionic variables allows for more variable manipulation under supersymmetric

transformations. Our choice of the action is defined below:

S0(X) =1

2(@h)2 and S1(X) = @2h (4.2)

S(X, 1, 2) =1

2(@h)2 � 1 2@2h (4.3)

where h is a real function of X, and @h = @h@x.

We can generate supersymmetric transformations for this action integral, which

are symmetries that exchange bosonic fields with fermionic fields and vice versa.

30

These will be generated using Grassmann odd variables, ✏1 and ✏2, and these anti-

commute with the fermionic variables. These transformations are given below:

�✏X = ✏1 1 + ✏2 2 (4.4)

� 1 = ✏2@h (4.5)

� 2 = �✏1@h (4.6)

To show that the action is invariant under these, we want �S = 0. This is done

using variational calculus as usual, but taking care of the negative sign in case of

swapping the Grassmann odd variables, and by taking the derivatives with respect

to the fermionic variables from the right.

Using the above defined action and it’s supersymmetric transformations, we will

exhibit the localization principle. For this purpose, we will split our work into seg-

ments, one that refers to points where @h 6= 0, and one that refers to all points in the

infinitesimal neighbourhood of @h = 0. For the first part, we will evoke the above

superysmmteric transformations to carry out a variable change such that one of the

fermionic variables goes to zero. This will simplify the path integral such that it goes

to zero. For the second part, we will look at all the fixed points of the fermionic

supersymmetric transformations, and write out the integral about the fixed points of

h to show that it reduces to a sum over a few points.

Suppose that @h 6= 0 everywhere. In this case, we can set the odd parameters as

follows:

✏1 = ✏2 = � 1

@h(4.7)

31

Plugging this substitution into the supersymmetric transformations, a change of vari-

able to S( bX, 0, b 2) is motivated as below: since S(X, 1, 2) = S( bX, 0, b 2)

bX = X + �✏X = X � 1

@h 2

b 1 = 1 + � 1 = 1 � 1 = 0

b 2 = 2 + � 2 = 2 + 1

The action should therefore remain invariant: S(X, 1, 2) = S( bX, 0, b 2)

This motivates us to adopt the following variable change which satisfies the above

rules:

bX = X � 1 2

@hb 1 = ↵(X) 1

b 2 = 2 + 1

where ↵(X) is an arbitrary function of X. We do not simply set b 1 = 0 as a variable

since this would trivialize our integration measure. In these variables, it is easy to

see that the integration measure is given by:

dXd 1d 2 = (↵( bX)� @2h( bX)

(@h( bX))2b 1 b 2)d bX d b 1 d b 2 (4.8)

The partition function using the action and the integration measure in these new

variables is given by:

Z =

Zd bX d b 1 d b 2↵( bX)e�S( bX,0, b 2) �

Z@2h( bX)

(@h( bX))2b 1 b 2d bX d b 1 d b 2e�S( bX,0, b 2)

(4.9)

32

The first term is 0 since b 1 does not appear in the integrand. The second term will

survive the Grassmann integration but can be expressed as a total derivative in bX.

Hence, if @h 6= 0, then the partition function is 0. For a more general case, we

consider @h to be 0 at some X’s. In this case, we cannot adopt the same change in

variable since bX is undefined for it. Here, we integrate the path integral piece-wise:

over all points where @h 6= 0 and therefore the contribution is to path integral is 0,

and then over all points in the infinitesimal neighbourhood of @h = 0, from where we

expect our non-zero contributions.

For points where @h = 0, both the fermionic supersymmetric transformations are

0. Hence, any contirbution to the partition function localized to the vicinity of the

fixed points of fermionic supersymmetrix transformations. This is a general principle

that we shall continue to apply to higher dimensions ass it holds in general for any

QFT with supersymmetry. Any contributions therefore are arising from the critical

points of the function h(X) (the stationary points, or the maxima or minima). Taking

h(X) to be any generic polynomial that is of order n, the number of (isolated) critical

points is at maximum n-1. We can taylor expand h(X) as a function about a critical

point as follows:

h(X) = h(Xc) +@2h(Xc)

2(X � Xc)

2 + ... (4.10)

Since our partition function will only localize about the critical points, we will only

consider the X in the very small neighbourhood around the critical points and can

therefore ignore higher order terms for those points. Then, the partition integral has

simplified to a summation over all the points in the neighbourhood of the critical

points of h(X). To express this, we first need to express S0(X) and S1(X) for these

33

points:

S0(X) =1

2(@h)2 =

1

2(@2h(Xc)(X �Xc))

2 =1

2↵2c(X �Xc)

2

S1(X) = @2h = ↵c

Then, the path integral can be reduced to a summation where the action uses the

above expressions, using the Grassmann integration rules and gaussian integration:

Z =

ZdX d 1 d 2 e�S0(X)+ 1 2S1(X)

=XXc

ZdX d 1 d 2

p2⇡

e�12↵c

2(X�Xc)2+↵c 1 2

=XXc

|↵c|↵c

=XXc

|@2h(Xc)|@2h(Xc)

This partition function is then basically adding a +1 for every minima in h(X) and

a -1 for every maxima in h(X), since @2h(Xc) is the term that is > 0 for minimum

stationary points and < 0 for maximum stationary points. If h is odd, then there are

an even number of isolated critical points, and half of them are even and half of them

are odd. If h is even, then we have an odd number of isolated critical points, and

there is a di↵erence of 1 between the number of maxima and minima points. Hence,

the partition function becomes:

Z = 0 if order of h is odd (4.11)

= ±1 if order of h is even (4.12)

4.2 Path integrals in 1-dimension

QFT can be formulated in one-dimensional space on the real line R, on a finite interval

I, or on a circle S1. The quantum field theory is generally parametrized by time t. For

34

the bosonic case, a single field is given by X(t). Then, an integral using the action,

over all paths connecting X(t1) = X1 and X(t2) = X2 can be given by: [4]

Z(X2, t2;X1, t1) =

ZDX(t)eiS(X) (4.13)

In general, this integral is summing up phases, but the convergence of the integral can

be a concern. For this purpose, we perform a Wick rotation on the time co-ordinate,

t� i⌧

where ⌧ is the Euclidean time, and the analogous Euclidean action using this time is

given by:

S =

Z{12(dX

dt)2 � V (X)}dt = i

Z{12(dX

d⌧)2 � V (X)}d⌧ = iSE(X)

Then, the integral is given by:

Z(X2, ⌧2;X1, ⌧1) =

ZDX(⌧)e�SE(X) (4.14)

And for the Euclidean path integral computed on the circle S1�, which is a circle of

circumference beta, the partition function can be written such that it is considered the

Euclidean path integral on an interval of length � with the initial and final endpoint

values of X being integrated over:

ZE(�) =

ZX(⌧+�)=X(⌧)

DX(⌧)e�SE(X) = Tre��H (4.15)

For the bosonic case, the partition function is given by: [5]

Z =

ZDXeiS(X) = Tr [e�iHt]

35

Using the results of bosonic and the fermionic cases that show the trace and their

equivalent path integral forms, the following results are computed on a circle of cir-

cumference � (Euclidean form).

Z(�) = Tr[e��H ] =

ZA

DX D D e�S(X, , )

Str[e��H ] = Tr[(�1)F e��H ] =

ZP

DX D D e�S(X, , )

The Witten index computed is in general unchanged under smooth deformations

of the theory, since the action integral is invariant under change of the pre-potential

h(x), with a constant term or upto a multiplicative constant, of course as long as the

constant takes large values so that at the ends the e�S(x) term is not infinite. We

can take the pre-potential as suggested and then the Hamiltonian takes the following

form:

h(x)h(x) where � >> 1 (4.16)

H =1

2p2 +

2

2(h0(x))2 +

2h00(x)[ , ] (4.17)

As �1, the potential term will go to infinity also. Then, the lowest energy states

should be centered around the smallest value of (h0(x))2, that is, they should be

centered around h0(x) = 0: the critical points of h(x). The deformation invariance

of the pre-potential gives the same result as the derived in the last section using the

localization principle.

36

4.3 Localization Principle in 1-D

The Witten index can be evaluated using the path integral:

Tr[(�1)F e��H ] =

ZP

DX D D e�S(X, , ) (4.18)

The Euclidean action and the supersymmetry transformations are given by: [4]

S =

Z 2⇡

0

{12(@x

@⌧)2 +

1

2(h0(x))2 +

@

@⌧+ h00(x) }d⌧ (4.19)

�x = ✏ � ✏ (4.20)

� = ✏(�@x@⌧

+ h0(x)) (4.21)

� = ✏(@x

@⌧+ h0(x)) (4.22)

These transformations are in keeping with the periodic fermionic and bosonic bound-

ary conditions. The invariance of the action has already been shown in the earlier

section in it’s non-euclidean form and the euclidean analog just requires using the

chain rule to obtain it in terms of ⌧ instead of t.

The proof used in zero-dimensional quantum field theory refers to a general princi-

ple prevalent in supersymmetry where if the action is invariant under supersymmetric

transformations, then for all @h(x) 6= 0, a suitable substitution of the parameter al-

lows us to change variables and set one of the new fermionic fields to zero, so that

the action is simplified and the integral goes to 0. The integral will localize to all

points where the supersymmetric variations of the fermionic fields will vanish. The

result can be argued using the same principles and the integration rules over fermions.

These principles apply in general to any quantum field theory that is supersymmetric,

37

in any dimension. For this case, the above fermionic fields go to zero when:

dx

d⌧= h0(x) = 0 (4.23)

that is, for the constant maps to the critical points of h(x). h(x) can again be expanded

about any critical point as follows:

h(x) = h(xi) +1

2h00(xi)(x� xi)

2 + ... (4.24)

Since we will only be looking at the infinitesimal neighbourhood of the critical points

for contributions to the path integral, the term (xi)(x�xi) is small and we can ignore

it’s higher order powers. The action integral for any general critical point is given by:

S(i) =

Z 2⇡

0

{12(x� xi)(� d2

d⌧ 2+ h00(xi)

2)(x� xi) + (d

d⌧+ h00(xi)) }d⌧ (4.25)

The path integral is given by:

Tr[(�1)F e��H ] =

ZP

D(x� xi) D D e�S(i) (4.26)

=det(@⌧ + h00(xi))pdet(�@2⌧ + (h00(xi))2)

(4.27)

=

Yn

(in + h00(xi))Yn

p(n2 + (h00(xi))2)

(4.28)

=h00(xi)

|h00(xi)| (4.29)

=NXi=1

sign(h00(xi)) (4.30)

Note that the periodic boundary condition for the fermions is crucial for the

existence of supersymmetry, as shown by Eq. (10.165), in the path- integral. If

we imposed anti-periodic boundary conditions there would be no supersymmetry to

38

begin with and our arguments about localization would not hold. This is the reason

the partition function without the insertion of (1)F (i.e., with anti-periodic boundary

conditions for fermions) does not localize near the critical points.

39

Chapter 5

Conclusion

The supersymmetric algebra of a generalized quantum field theory is given as follows:

{Q, Q†} = 2H

{Q, Q} = 0 , {Q†, Q†} = 0

[Q, H] = 0 , [Q†, H]

This along with the more detailed algebra expressed above shows that any state | 0iwith energy E=0, is necessarily a ground state and supersymmetric: it is invariant

under the action of these supercharges. This also exhibits a degeneracy in the energy

levels for E 6= 0, splitting the Hilbert space evenly into the bosonic and fermionic

sub-spaces.

The Witten Index is a quantity defined to describe the ground state of a supersym-

metric field theory, by counting the di↵erence in the number of bosonic and fermionic

ground states. It is invariant under smooth deformations. If there are no states with

E = 0, then the Witten Index is 0 and supersymmetry is spontaneously broken. If

the Witten index is non-zero, supersymmetry cannot be spontaneously broken: there

40

always exists at least one ground state with E = 0. Based on the value of the Witten

index, +1 or �1, this ground state is bosonic or fermionic in nature.

The Witten Index is also computed in a path integral formalism using the local-

ization principle: the contribution to the partition function is localized to the vicinity

of the fixed points of fermionic supersymmetric transformations. These are mapped

by the critical points of the pre-potential. This is a feature typical of of any general

supersymmetric quantum theory, and holds for higher dimensions. When applied

to the 1-dimensional case, the partition function (equivalent to the Witten Index)

becomes:

Z = 0 if order of h is odd

= ±1 if order of h is even

This thesis introduces a generalized supersymmetric quantum field theory and

shows the importance of the localization principle in solving path integrals that de-

scribe the ground states of such systems. A useful application of this is in supersym-

metric quantum field theory in 1+1 dimension where the path integrals can not be

evaluated otherwise.

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Bibliography

[1] Bijan Kumar Bagchi. Supersymmetry in Quantum and Classical Mechanics, chap-ter 2, 3, 4. Chapman Hall/CRC, 2001.

[2] Arthur Boetes. An Introduction to Supersymmetry in Quantum Mechanical Sys-tems. Master’s thesis, University of Amsterdam, October 2011.

[3] Fiorenzo Bastianelli. Path Integrals for Fermions and Supersymmetric QuantumMechanics. http://www-th.bo.infn.it/people/bastianelli/ch3-FT2.pdf,May 2015.

[4] Albrecht Klemm Kentaro Hori, Sheldon Katz. Mirror Symmetry, chapter 9, 10.American Mathematical Society for Clay Mathematics Institute, 2003.

[5] Mark Srednicki. Quantum Field Theory. http://www.physics.ucsb.edu/mark/qft.html,, September 2006.

[6] T Wellman. An Introduction to Supersymmetry in Quantum Mechanical Systems.Master’s thesis, Brown University, April 2003.

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