![2 Field Theory: A Path Integral Approach
1.2 Metric and Other Notations
In these lectures, we will discuss both non-relativistic as well as rel-
ativists theories. For the relativistic case, we will use the Bjorken-
Drell convention. Namely, the contravariant coordinates are assumed
to be
x» = (t,x), /i = 0,l,2,3, (1.1)
while the covariant coordinates have the form
X/j, = T],MVX'/
= (t, - x ) . (1.2)
Here we have assumed the speed of light to be unity (c = 1). The
covariant metric, therefore, follows to have a diagonal form with the
signatures
Vnu = ( + , - , - , - ) • (1.3)
The inverse or the contravariant metric clearly also has the same
form, namely,
TT = ( + , - , - , - ) • (1.4)
The invariant length is given by
x2
= x% = rTxpXv = ri^x" = t2
- x2
. (1.5)
The gradients are similarly obtained from Eqs. (1.1) and (1.2) to be
a
-=aMf'v
)' (L6)
<"=£=(*•-*)• <17)
so that the D'Alembertian takes the form
• = d^ = rrd»dv = ~-V2
. (1.8)](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-21-320.jpg)
![Introduction 3
1.3 Functionals
It is evident that in dealing with dynamical systems, we are dealing
with functions of continuous variables. In fact, most of the times,
we are really dealing with functions of functions which are otherwise
known as functionals. If we are considering the motion of a particle
in a potential in one dimension, then the Lagrangian is given by
L(x, x) = - m i 2
- V(x), (1.9)
where x(t) and x(t) denote the coordinate and the velocity of the
particle and the simplest functional we can think of is the action
functional defined as
S[x] = [ f
dtL(x,x). (1.10)
Note that unlike a function whose value depends on a particular
point in the coordinate space, the value of the action depends on
the entire trajectory along which the integration is carried out. For
different paths connecting the initial and the final points, the value
of the action functional will be different.
Thus, a functional has the generic form
Ff] = JdxF(f(x)), (1.11)
where, for example, we may have
F(f(x)) = (f(x))n
. (1.12)
Sometimes, one loosely also says that F(f(x)) is a functional. The
notion of a derivative can be extended to the case of functionals in a
natural way through the notion of generalized functions. Thus, one
defines the functional derivative or the Gateaux derivative from the
linear functional
F'v]=^-F{f + ev] =
/ d l
^ " W
- ( U 3 )](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-22-320.jpg)
![4 Field Theory: A Path Integral Approach
Equivalently, from the working point of view, this simply corresponds
to defining
Sf(y) e - o €
l
' '
It now follows from Eq. (1.14) that
Sf{x)
Sf(y)
8(x-y). (1.15)
The functional derivative satisfies all the properties of a deriva-
tive, namely, it is linear and associative,
5
•(FM+FM=m+mm
5f(XyllJi ZUiJ
5f{x) 6f(x)'
5
<F1[f]F2[f]) = 5
-^4F2[f} + F1[f}S
^4. (1.16)
6f{xy xWJ ,uu Sf{x) . W J x W J 5f{x)
It also satisfies the chain rule of differentiation. Furthermore, we now
see that given a functional F[f], we can Taylor expand it in the form
F[f] = fdx P0(x) + fdXldx2 Pi(xux2) f(x2)
+ dx1dx2dx3P2(xi,x2,x3) f(x2)f(x3)- , (1.17)
where
Pi(xi,x2) =
P2(xi,x2,x3) =
P0{x) = F(f(x))nx)=0 ,
8F(f(xi))
Sf(x2)
1 PFifixt))
(1.18)
f(x)=0
2! 5f(x2)8f(x3) /(x)=0
and so on.
As simple examples, let us calculate a few particular functional
derivatives.](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-23-320.jpg)
![Introduction 5
(i) Let
F[f] = Jdy F(f(y)) = J dy (f(y))n
, (1.19)
where n denotes a positive integer. Then,
SF(f(y)) = H m F(f(y) + e5(y-x))-F(f(y))
Sf(x) e™ e
l]m(f(y) + e5(y-x))n
-(f(y))n
= H m ( / ( y ) ) n + n
<^y))n
~l
^y - *) + °(e2
) - (f(y»n
= n ( / ( y ) ) n _ 1
% - a ; ) . (1.20)
Therefore, we obtain
SFf] _ f 5F(f(y))
6f(x) J V
8f(x)
= Jdyn(f(y)r-1
5(y-x)
= n(f(x))n
~1
. (1.21)
(ii) Let us next consider the one-dimensional action in Eq. (1.10)
S[x] = [ f
dt' L(x(t'),x(t')), (1.22)
with
L(x(t),x(t)) = m{x{t)f -V(x(t))
= T{x(t))-V(x(t)). (1.23)
In a straightforward manner, we obtain
SV(x(t')) = y(s(f) + e£ft' - t)) - V{x(t'))
Sx(t) e^o e
= ^'(x^'))*^ - *), (1-24)](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-24-320.jpg)
![6 Field Theory: A Path Integral Approach
where we have defined
Similarly,
ST(x(t')) = U m T(x(t>) + e£r5(t>-t))-T(x(t>))
5x(t) e^O €
= mx(t')—5(t'-t). (1.25)
It is clear now that
SL(x(t'),x(t')) _ 5{T{x(t'))-V{x{t')))
8x(t) ~ Sx(t)
= mx(t') -^6(1/ -t)- V'{x(t'))5{t' - i). (1.26)
Consequently, in this case, we obtain for ti <t <tf
5S[x] = /•*/ 5L(x(t'),x(t'))
8x(t) Jti 6x(t)
= ff
dt' {mx{t')^-/8(t' -t)- V'(x{t'))5{t' - t))
Jti dt
= -mx(t) - V'(x(t))
ddL(x(t),x(t)) dL(x(t),x(t))
dt dx{t) +
dx(t) ' {
' '
The right hand side is, of course, reminiscent of the Euler-Lagrange
equation. In fact, we note that
6x(t) dt 8x(t) 8x{t) ' K
' '
gives the Euler-Lagrange equation as a functional extremum of the
action. This is nothing other than the principle of least action ex-
pressed in a compact notation in the language of functionals.](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-25-320.jpg)
![Path Integrals and Quantum Mechanics 15
Here we have used the completeness relations of the momentum basis
states given in Eq. (2.4) as well as the defining relations in Eqs. (2.1),
(2.3) and (2.5). (The matrix element of the quantum Hamiltonian is a
classical function for which the ordering is irrelevant.) To understand
Weyl ordering, on the other hand, let us note that the expansion of
(axop + j3pop)N
,
generates the Weyl ordering of products of the form a^opPop naturally
if we treat xop and pop as non-commuting operators. In fact, we can
easily show that
(*xop + (3pop)N
= ] T - ^ an
r(xn
opP™)W
-°- (2.14)
n+m=N
The expansion of the exponential operator
e(aa;op+/3pop)
would, of course, generate all such powers and by analyzing the ma-
trix elements of this exponential operator, we will learn about the
matrix elements of Weyl ordered Hamiltonians.
Prom the fact that the commutator of a;op and pop is a constant,
we obtain using the Baker-Campbell-Hausdorff formula
Q 2 )ePPope( 2 I = gl 2 ) (APPop~ 2 ~ >
— e(ax0p+/3p0p) _ (2.15)
Using this relation, it can now be easily shown that
dp (x e(
2 >epp
°p
p){peK
2 >x)
/
p-e-fc*-* C-^+0v). (2.16)](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-34-320.jpg)
![16 Field Theory: A Path Integral Approach
Once again, we have used here the completeness properties given in
Eq. (2.4) as well as the defining relations in Eqs. (2.1), (2.3) and (2.5).
It follows from this that for a Weyl ordered quantum Hamiltonian,
we will have
< z ' | # w
- ° - ( W o p ) | , ) = f ^ e-i«°-*H {^f,p) • (2.17)
As we see, the matrix elements of the Weyl ordered Hamiltonian leads
to what is known as the mid-point prescription and this is what we
will use in all of our discussions.
We are now ready to calculate the transition amplitude. Let us
recall that in the Heisenberg picture, for tf > U, we have
U (tf,Xf]ti,Xi) = H(Xf,tfXi,ti)H.
Let us divide the time interval between the initial and the final time
into N equal segments of infinitesimal length e. Namely, let
* = tj
jr • (2.i8)
In other words, for simplicity, we discretize the time interval and
in the end, we are interested in taking the continuum limit e —
> 0
and N —
> oo such that Eq. (2.18) holds true. We can now label the
intermediate times as, say,
tn = U + ne, n = l , 2 , . . . , ( i V - l ) . (2.19)
Introducing complete sets of coordinate basis states for every in-
termediate time point (see Eq. (2.9)), we obtain
U(tf,xf,ti,Xi) = H(xf,tfxi,ti)H
= lim dx1---dxN-lH{xf,tfxN-i,tN-i)H
N-^oo
X H{XN-l-,tN-lxN-2,tN-2)H • • -H {xi,hXi,ti)H .
(2.20)](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-35-320.jpg)
![Path Integrals and Quantum Mechanics 19
" / 2nE e
m(xn-xn_i)ji
( m(xn-xn_i) '
'-)-(•-
1 (2'Kmh2
gif1
""1
"-1
= ( — Y
2mheJ
(2.27)
Substituting this back into the transition amplitude in Eq. (2.26),
we obtain
N_
U(tf,xf;tl,xi) = lim ( ^ ) 2
/
iey^AT I m fx
n-x
n-l y(Xn+x
n-l
dxi • • • dxN-ieh n _ 1
V 2
^ e
' ^
= A f Vx etti<l
*m±2
-V
^
= AJvxe*s
W, (2.28)
where A is a constant independent of the dynamics of the system
and S[x] is the action for the system given in Eq. (1.10). This is
Feynman's path integral for the transition amplitude in quantum
mechanics.](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-38-320.jpg)
![Path Integrals and Quantum Mechanics 23
path integral representation of the transition amplitude. Let us recall
that the Schrodinger equation is a differential equation. Therefore, it
determines infinitesimal changes in the wave function. Consequently,
to derive the Schrodinger equation, we merely have to examine the
infinitesimal form of the transition amplitude or the path integral.
Prom the explicit form of the transition amplitude in Eq. (2.29), we
obtain for infinitesimal e
U(tf = e,xf,ti = 0,Xi)
/m^Kl^'-vpP)). (M1)
2-Kihe) v
'
We also know from Eq. (1.38) that the transition amplitude is the
propagator which gives the propagation of the wave function in the
following way,
/
oo
dx' U(e,x;0,x')il>(x',0). (2.32)
-oo
Therefore, substituting the form of the transition amplitude namely,
Eq. (2.31) into Eq. (2.32), we obtain
1>(x, e) = ( ^ | - ) * | ° ° Me&l*-*'^f V
^H{x 0). (2.33)
Let us next change variables to
r] = x' - x , (2.34)
so that we can write
rP(x, e) = ( ~ ) h
J°° dV e S ^ - ^ ^ + l ) ] ^ , +v, 0). (2.35)
It is obvious that because e is infinitesimal, if 7
7 is large, then the first
term in the exponent would lead to rapid oscillations and all such
contributions will average out to zero. The dominant contribution
will, therefore, come from the region of integration
0 < H < ( ^ ) (2.36)](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-42-320.jpg)
![26 Field Theory: A Path Integral Approach
Therefore, from our definition of the transition amplitude in
Eq. (2.28) or (2.29), we obtain
U(tf,xf,ti,Xi)
= lim I ——-) / dxi • • • d ^ - i e h
^n = 1 2
^ £
)
e^o V 2mne I J
JV—>oo
lim ( ) 2
/ dxi • • •dxN-ie^'^n
=l(
-Xn
~Xn
e->o v 2-nine J J
i ) 2
TV—>oo
(2.42)
Defining
/ m
2heJ
we have
^ = U i d * « , (2-43)
U(tf,xf;U,Xi)= hm ( _ ) ^ _ J
—>oo
/"dyi • • -dj/jv-i e*E^=i(i/n-vn-i)a
. (2.44)
iV->oo
X
This is a Gaussian integral which can be evaluated in many different
ways. However, the simplest method probably is to work out a few
lower order ones and derive a pattern. We note that
[dyx e*[(w-i»)2
+(i«-yi)a
] = [dyi eiPfoi-^T^+^w-w)2
]
™*etto-vo) ( 2 .4 5 )](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-45-320.jpg)
![Path Integrals and Quantum Mechanics 27
If we had two intermediate integrations, then we will have
/ dyidy2 ei
[^1
~yo
^2+
^2
~?/1
^+
^3
'"?/2
^2
]
= (*I* fdy2eih(y2-y0)2
+(y3-y2?]
) ' /
d y 2 e ifte-^)2
+t(«)2
2 / V 3 '
(wr
i
, 2 2
ef(»/3-!/o) (2.46)
A pattern is now obvious and using this we can write
U(tf,xf,ti,Xi)
e-+o 2irihe) m N
e
JV—>oo
h
m f^-)f
f^V
_l
e-^o 2mne/ m I fl
e
N—*oo
e2HNc (xN—xo)
= h m ( ) e^hNeix
f x
i)
e^o 2mhNeJ
AT—>oo
m
2irih(tf — ti)
I i m
(x
f-x
i)
a
(*/-*0 (2.47)
Thus, we see that for a free particle, the transition amplitude can be
explicitly evaluated. It has the right behavior in the sense that, we
see as tf —> U,
U{tf,xf,ti,Xi)->8(xf-Xi), (2.48)](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-46-320.jpg)
![28 Field Theory: A Path Integral Approach
which is nothing other than the orthonormality relation for the
states in the Heisenberg picture given in Eq. (2.8). Second, all the
potentially dangerous singular terms involving e have disappeared.
Furthermore this is exactly what one would obtain by solving the
Schrodinger equation. It expresses the well known fact that even
a well localized wave packet spreads with time. That is, even the
simplest of equations has only dispersive solutions.
Let us note here that since
S[x] = dt -mi;2
,
J ti
the Euler-Lagrange equations give (see Eq. (1.28))
This gives as solutions
Xcl(t) = v = constant. (2.50)
Thus, for the classical trajectory, we have
S[xd] = f f
dt ^mx = ^mv2
(tf - U). (2.51)
On the other hand, since v is a constant, we can write
Xf - Xi = V(tf - ti)
or, v=X
f^. (2.52)
Substituting this back into Eq. (2.51), we obtain
*M=Mt^)'<''-«-=^. (2
-53)
We recognize, therefore, that we can also write the quantum transi-
tion amplitude, in this case, simply as](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-47-320.jpg)
![Path Integrals and Quantum Mechanics 29
This is a particular characteristic of some quantum systems which
can be exactly solved. Namely, for these systems, the transition
amplitude can be written in the form
U(tf,xf;ti,xi) = Ae^x
^, (2.55)
where A is a constant.
Finally, let us note from the explicit form of the transition ampli-
tude in Eq. (2.47) that
dU _ U im /xf — Xi
~dff ~ ~2{tf -U) ~~2htf-Uj
dU _ im fxf — %i TT
2 / „ 2
dxf h tf — t
d2
U _ im U (im fxf — Xi .
dxi h tf — ti h J tf — t
- J2
^ ( h U
4- — fXf
~Xi
2
Tf]
- V in
2{tf-U)+
2 t f - u ) )
2m {.BU . „ „ .
Therefore, it follows that
..8U ti2
d2
U , „ „ .
xh
wr-^!*f> (
2
-5
7
)
which is equivalent to saying that the transition amplitude obtained
from Feynman's path integral, indeed, solves the Schrodinger equa-
tion for a free particle (compare with Eq. (1.35)).](https://image.slidesharecdn.com/68492-250316203747-0f48645b/85/Field-theory-a-path-integral-approach-2nd-Edition-Ashok-Das-48-320.jpg)
Field theory a path integral approach 2nd Edition Ashok Das
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- 5. World Scientific Lecture Notes in Physics - Vol. 75 FIELD THEORY A Path Integral Approa, Second Edition ASHOK DAS
- 6. FIELD THEORY A Path Integral Approach Second Edition
- 7. World Scientific Lecture Notes in Physics Published Vol. 54: Introduction to the Theory of Spin Glasses and Neural Networks V Dotsenko Vol. 55: Lectures in Particle Physics D Green Vol. 56: Chaos and Gauge Field Theory T S Biro, et al. Vol. 57: Foundations of Quantum Chromodynamics (2nd ed.): An Introduction to Perturbative Methods in Gauge Theories TMuta Vol. 58: Concepts in Solids, Lectures on the Theory of Solids P. W. Anderson and H. Bunke Vol. 59: Lattice Gauge Theories: An Introduction (2nd ed.) H J Ftothe Vol. 60: Massive Neutrinos in Physics and Astrophysics (2nd ed.) R N Mohapatra and P B Pal Vol. 61: Modern Differential Geometry for Physicists (2nd ed.) C J Isham Vol. 62: ITEP Lectures on Particle Physics and Field Theory (In 2 Volumes) M A Shifman Vol. 64: Fluctuations and Localization in Mesoscopic Electron Systems M Janssen Vol. 65: Universal Fluctuations: The Phenomenology of Hadronic Matter R Sorer and M Ploszajczak Vol. 66: Microcanonical Thermodynamics: Phase Transitions in "Small" Systems DHE Gross Vol. 67: Quantum Scaling in Many-Body Systems M A Continentino Vol. 69: Deparametrization and Path Integral Quantization of Cosmological Models C Simeone Vol. 70: Noise Sustained Patterns: Fluctuations & Nonlinearities Markus Loecher Vol. 71: The QCD Vacuum, Hadrons and Superdense Matter (2nd ed.) Edward V Shuryak Vol. 72: Massive Neutrinos in Physics and Astrophysics (3rd ed.) R Mohapatra and P B Pal Vol. 73: The Elementary Process of Bremsstrahlung W Nakel and E Haug Vol. 74: Lattice Gauge Theories: An Introduction (3rd ed.) HJRothe
- 8. World Scientific Lecture Notes in Physics - Vol. 75 FIELD THEORY A Path Integral Approach Second Edition XV^ilUJx. JL/xk3 University of Rochester, USA i|p World Scientific NEW JERSEY > LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
- 9. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FIELD THEORY: A PATH INTEGRAL APPROACH Second Edition Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-847-6 ISBN 981-256-848-4 (pbk) Printed in Singapore by B & JO Enterprise
- 12. Preface to the First Edition Traditionally, field theory had its main thrust of development in high energy physics. Consequently, the conventional field theory courses are taught with a heavy emphasis on high energy physics. Over the years, however, it has become quite clear that the methods and tech- niques of field theory are widely applicable in many areas of physics. The canonical quantization methods, which is how conventional field theory courses are taught, do not bring out this feature of field the- ory. A path integral description of field theory is the appropriate setting for this. It is with this goal in mind, namely, to make gradu- ate students aware of the applicability of the field theoretic methods to various areas, that the Department of Physics and Astronomy at the University of Rochester introduced a new one semester course on field theory in Fall 1991. This course was aimed at second year graduate students who had already taken a one year course on nonrelativistic quantum mechan- ics but had not necessarily specialized into any area of physics and these lecture notes grew out of this course which I taught. In fact, the lecture notes are identical to what was covered in the class. Even in the published form, I have endeavored to keep as much of the de- tailed derivations of various results as I could — the idea being that a reader can then concentrate on the logical development of concepts without worrying about the technical details. Most of the concepts were developed within the context of quantum mechanics — which the students were expected to be familiar with — and subsequently these concepts were applied to various branches of physics. In writ- ing these lecture notes, I have added some references at the end of vii
- 13. viii Field Theory: A Path Integral Approach every chapter. They are only intended to be suggestive. There is so much literature that is available in this subject that it would have been impossible to include all of them. The references are not meant to be complete and I apologize to many whose works I have not cited in the references. Since this was developed as a course for general students, the many interesting topics of gauge theories are also not covered in these lectures. It simply would have been impossible to do justice to these topics within a one semester course. There are many who were responsible for these lecture notes. I would like to thank our chairman, Paul Slattery, for asking me to teach and design a syllabus for this course. The students deserve the most credit for keeping all the derivations complete and raising many issues which I, otherwise, would have taken for granted. I am grateful to my students Paulo Bedaque and Wen-Jui Huang as well as to Dr. Zhu Yang for straightening out many little details which were essential in presenting the material in a coherent and consistent way. I would also like to thank Michael Begel for helping out in numerous ways, in particular, in computer-generating all the figures in the book. The support of many colleagues was also vital for the completion of these lecture notes. Judy Mack, as always, has done a superb job as far as the appearance of the book is concerned and I sincerely thank her. Finally, I am grateful to Ammani for being there. Ashok Das, Rochester.
- 14. Preface to the Second Edition This second edition of the book is an expanded version which con- tains a chapter on path integral quantization of gauge theories as well as a chapter on anomalies. In addition, chapter 6 (Supersymmetry) has been expanded to include a section on supersymmetric singular potentials. While these topics were not covered in the original course on path integrals, they are part of my lectures in other courses that I have taught at the University of Rochester and have been incorpo- rated into this new edition at the request of colleagues from all over the world. There are many people who have helped me to complete this edition of the book and I would like to thank, in particular, Judy Mack, Arsen Melikyan, Dave Munson and J. Boersma for all their assistance. Ashok Das, Rochester.
- 16. Contents Preface to the First Edition vii Preface to the Second Edition ix 1. Introduction 1 1.1 Particles and Fields 1 1.2 Metric and Other Notations 2 1.3 Functionals 3 1.4 Review of Quantum Mechanics 7 1.5 References 10 2. Path Integrals and Quantum Mechanics 11 2.1 Basis States 11 2.2 Operator Ordering 13 2.3 The Classical Limit 20 2.4 Equivalence with the Schrodinger Equation 22 2.5 Free Particle 25 2.6 References 30 3. Harmonic Oscillator 31 3.1 Path Integral for the Harmonic Oscillator 31 3.2 Method of Fourier Transform 33 3.3 Matrix Method 36 3.4 The Classical Action 45 xi
- 17. xii Field Theory: A Path Integral Approach 3.5 References 51 4. Generating Functional 53 4.1 Euclidean Rotation 53 4.2 Time Ordered Correlation Functions 59 4.3 Correlation Functions in Definite States 61 4.4 Vacuum Functional 64 4.5 Anharmonic Oscillator 71 4.6 References 73 5. Path Integrals for Fermions 75 5.1 Fermionic Oscillator 75 5.2 Grassmann Variables 78 5.3 Generating Functional 83 5.4 Feynman Propagator 86 5.5 The Fermion Determinant 91 5.6 References 95 6. Supersymmetry 97 6.1 Supersymmetric Oscillator 97 6.2 Supersymmetric Quantum Mechanics 102 6.3 Shape Invariance 105 6.4 Example 110 6.5 Supersymmetry and Singular Potentials Ill 6.5.1 Regularized Superpotential 115 6.5.2 Alternate Regularization 117 6.6 References 118 7. Semi-Classical Methods 121 7.1 WKB Approximation 121 7.2 Saddle Point Method 127 7.3 Semi-Classical Methods in Path Integrals 130 7.4 Double Well Potential 134 7.5 References 142
- 18. Contents xiii 8. Path Integral for the Double Well 143 8.1 Instantons 143 8.2 Zero Modes 150 8.3 The Instanton Integral 154 8.4 Evaluating the Determinant 158 8.5 Multi-Instanton Contributions 163 8.6 References 166 9. Path Integral for Relativistic Theories 167 9.1 Systems with Many Degrees of Freedom 167 9.2 Relativistic Scalar Field Theory 170 9.3 Feynman Rules 181 9.4 Connected Diagrams 184 9.5 References 186 10. Effective Action 187 10.1 The Classical Field 187 10.2 Effective Action 193 10.3 Loop Expansion 200 10.4 Effective Potential at One Loop 203 10.5 References 208 11. Invariances and Their Consequences 209 11.1 Symmetries of the Action 209 11.2 Noether's Theorem 212 11.2.1 Example 215 11.3 Complex Scalar Field 218 11.4 Ward Identities 222 11.5 Spontaneous Symmetry Breaking 226 11.6 Goldstone Theorem 235 11.7 References 236 12. Gauge Theories 239 12.1 Maxwell Theory 239 12.2 Non-Abelian Gauge Theory 246 12.3 Path Integral for Gauge Theories 255
- 19. xiv Field Theory: A Path Integral Approach 12.4 BRST Invariance 266 12.5 Ward Identities 274 12.6 References 278 13. Anomalies 279 13.1 Anomalous Ward Identity 279 13.2 Schwinger Model 289 13.3 References 307 14. Systems at Finite Temperature 309 14.1 Statistical Mechanics 309 14.2 Critical Exponents 314 14.3 Harmonic Oscillator 318 14.4 Fermionic Oscillator 324 14.5 References 326 15. Ising Model 327 15.1 One Dimensional Ising Model 327 15.2 The Partition Function 332 15.3 Two Dimensional Ising Model 337 15.4 Duality 339 15.5 High and Low Temperature Expansions 343 15.6 Quantum Mechanical Model 349 15.7 Duality in the Quantum System 356 15.8 References 358 Index 359
- 20. Chapter 1 Introduction 1.1 Particles and Fields Classically, there are two kinds of dynamical systems that we en- counter. First, there is the motion of a particle or a rigid body (with a finite number of degrees of freedom) which can be described by a finite number of coordinates. And then, there are physical systems where the number of degrees of freedom is nondenumerably (non- countably) infinite. Such systems are described by fields. Familiar examples of classical fields are the electromagnetic fields described by E(x, t) and B(x, t) or equivalently by the potentials (0(x, t), A(x, t)). Similarly, the motion of a one-dimensional string is also described by a field 4>(x,t), namely, the displacement field. Thus, while the coor- dinates of a particle depend only on time, fields depend continuously on some space variables as well. Therefore, a theory described by fields is usually known as a D + 1 dimensional field theory where D represents the number of spatial dimensions on which the field vari- ables depend. For example, a theory describing the displacements of the one-dimensional string would constitute a 1+1 dimensional field theory whereas the more familiar Maxwell's equations (in four dimensions) can be regarded as a 3+1 dimensional field theory. In this language, then, it is clear that a theory describing the motion of a particle can be regarded as a special case, namely, we can think of such a theory as a 0+1 dimensional field theory. 1
- 21. 2 Field Theory: A Path Integral Approach 1.2 Metric and Other Notations In these lectures, we will discuss both non-relativistic as well as rel- ativists theories. For the relativistic case, we will use the Bjorken- Drell convention. Namely, the contravariant coordinates are assumed to be x» = (t,x), /i = 0,l,2,3, (1.1) while the covariant coordinates have the form X/j, = T],MVX'/ = (t, - x ) . (1.2) Here we have assumed the speed of light to be unity (c = 1). The covariant metric, therefore, follows to have a diagonal form with the signatures Vnu = ( + , - , - , - ) • (1.3) The inverse or the contravariant metric clearly also has the same form, namely, TT = ( + , - , - , - ) • (1.4) The invariant length is given by x2 = x% = rTxpXv = ri^x" = t2 - x2 . (1.5) The gradients are similarly obtained from Eqs. (1.1) and (1.2) to be a -=aMf'v )' (L6) <"=£=(*•-*)• <17) so that the D'Alembertian takes the form • = d^ = rrd»dv = ~-V2 . (1.8)
- 22. Introduction 3 1.3 Functionals It is evident that in dealing with dynamical systems, we are dealing with functions of continuous variables. In fact, most of the times, we are really dealing with functions of functions which are otherwise known as functionals. If we are considering the motion of a particle in a potential in one dimension, then the Lagrangian is given by L(x, x) = - m i 2 - V(x), (1.9) where x(t) and x(t) denote the coordinate and the velocity of the particle and the simplest functional we can think of is the action functional defined as S[x] = [ f dtL(x,x). (1.10) Note that unlike a function whose value depends on a particular point in the coordinate space, the value of the action depends on the entire trajectory along which the integration is carried out. For different paths connecting the initial and the final points, the value of the action functional will be different. Thus, a functional has the generic form Ff] = JdxF(f(x)), (1.11) where, for example, we may have F(f(x)) = (f(x))n . (1.12) Sometimes, one loosely also says that F(f(x)) is a functional. The notion of a derivative can be extended to the case of functionals in a natural way through the notion of generalized functions. Thus, one defines the functional derivative or the Gateaux derivative from the linear functional F'v]=^-F{f + ev] = / d l ^ " W - ( U 3 )
- 23. 4 Field Theory: A Path Integral Approach Equivalently, from the working point of view, this simply corresponds to defining Sf(y) e - o € l ' ' It now follows from Eq. (1.14) that Sf{x) Sf(y) 8(x-y). (1.15) The functional derivative satisfies all the properties of a deriva- tive, namely, it is linear and associative, 5 •(FM+FM=m+mm 5f(XyllJi ZUiJ 5f{x) 6f(x)' 5 <F1[f]F2[f]) = 5 -^4F2[f} + F1[f}S ^4. (1.16) 6f{xy xWJ ,uu Sf{x) . W J x W J 5f{x) It also satisfies the chain rule of differentiation. Furthermore, we now see that given a functional F[f], we can Taylor expand it in the form F[f] = fdx P0(x) + fdXldx2 Pi(xux2) f(x2) + dx1dx2dx3P2(xi,x2,x3) f(x2)f(x3)- , (1.17) where Pi(xi,x2) = P2(xi,x2,x3) = P0{x) = F(f(x))nx)=0 , 8F(f(xi)) Sf(x2) 1 PFifixt)) (1.18) f(x)=0 2! 5f(x2)8f(x3) /(x)=0 and so on. As simple examples, let us calculate a few particular functional derivatives.
- 24. Introduction 5 (i) Let F[f] = Jdy F(f(y)) = J dy (f(y))n , (1.19) where n denotes a positive integer. Then, SF(f(y)) = H m F(f(y) + e5(y-x))-F(f(y)) Sf(x) e™ e l]m(f(y) + e5(y-x))n -(f(y))n = H m ( / ( y ) ) n + n <^y))n ~l ^y - *) + °(e2 ) - (f(y»n = n ( / ( y ) ) n _ 1 % - a ; ) . (1.20) Therefore, we obtain SFf] _ f 5F(f(y)) 6f(x) J V 8f(x) = Jdyn(f(y)r-1 5(y-x) = n(f(x))n ~1 . (1.21) (ii) Let us next consider the one-dimensional action in Eq. (1.10) S[x] = [ f dt' L(x(t'),x(t')), (1.22) with L(x(t),x(t)) = m{x{t)f -V(x(t)) = T{x(t))-V(x(t)). (1.23) In a straightforward manner, we obtain SV(x(t')) = y(s(f) + e£ft' - t)) - V{x(t')) Sx(t) e^o e = ^'(x^'))*^ - *), (1-24)
- 25. 6 Field Theory: A Path Integral Approach where we have defined Similarly, ST(x(t')) = U m T(x(t>) + e£r5(t>-t))-T(x(t>)) 5x(t) e^O € = mx(t')—5(t'-t). (1.25) It is clear now that SL(x(t'),x(t')) _ 5{T{x(t'))-V{x{t'))) 8x(t) ~ Sx(t) = mx(t') -^6(1/ -t)- V'{x(t'))5{t' - i). (1.26) Consequently, in this case, we obtain for ti <t <tf 5S[x] = /•*/ 5L(x(t'),x(t')) 8x(t) Jti 6x(t) = ff dt' {mx{t')^-/8(t' -t)- V'(x{t'))5{t' - t)) Jti dt = -mx(t) - V'(x(t)) ddL(x(t),x(t)) dL(x(t),x(t)) dt dx{t) + dx(t) ' { ' ' The right hand side is, of course, reminiscent of the Euler-Lagrange equation. In fact, we note that 6x(t) dt 8x(t) 8x{t) ' K ' ' gives the Euler-Lagrange equation as a functional extremum of the action. This is nothing other than the principle of least action ex- pressed in a compact notation in the language of functionals.
- 26. Introduction 7 1.4 Review of Quantum Mechanics In this section, we will describe very briefly the essential features of quantum mechanics assuming that the readers are familiar with the subject. The conventional approach to quantum mechanics starts with the Hamiltonian formulation of classical mechanics and pro- motes observables to non-commuting operators. The dynamics, in this case, is given by the time-dependent Schrodinger equation * * ! * £ » _ H h W > , ( 1 3 ) where H denotes the Hamiltonian operator of the system. Equiva- lently, in the one dimensional case, the wave function of a particle satisfies h2 d2 + V(x))^(x,t), (1.30) 2m dx2 where we have identified # E , * ) = <*#(*)>. (1-31) with x) denoting the coordinate basis states. This, then, defines the time evolution of the system. The main purpose behind solving the Schrodinger equation lies in determining the time evolution operator which generates the time translation of the system. Namely, the time evolution operator trans- forms the quantum mechanical state at an earlier time ti to a future time t as M*l)) = tf(*l,*2M*2)>. (1-32) Clearly, for a time independent Hamiltonian, we see from Eq. (1.29) (the Schrodinger equation) that for t > t2, U(h,t2) = e-^tl -t2)H . (1.33)
- 27. 8 Field Theory: A Path Integral Approach More explicitly, we can write U(h,t2) = B(h - t2)e-^-^H . (1.34) It is obvious that the time evolution operator is nothing other than the Green's function for the time dependent Schrodinger equation and satisfies (ih-^- - H U(h,t2) = ih5(h - *2) • (1-35) Determining this operator is equivalent to finding its matrix elements in a given basis. Thus, for example, in the coordinate basis defined by Xx)=xx), (1.36) we can write {xiU(t1,t2)x2) = U(ti,x1;t2,x2). (1-37) If we know the function U(ti,xit2,x2) completely, then the time evolution of the wave function can be written as ip(xi,h) = dx2U{ti,xi;t2,x2)ip(x2,t2). (1.38) It is interesting to note that the dependence on the intermediate times drops out in the above equation as can be easily checked. Our discussion has been within the framework of the Schr-odinger picture so far where the quantum states i/j(t)) carry time dependence while the operators are time independent. On the other hand, in the Heisenberg picture, where the quantum states are time independent, using Eq. (1.32) we can identify IV>H = m = o))s = m = o)) (1.39)
- 28. Introduction 9 In this picture, the operators carry all the time dependence. For example, the coordinate operator in the Heisenberg picture is related to the coordinate operator in the Schrodinger picture through the relation XH{t) = e*tH Xe-*tH . (1.40) The eigenstates of this operator satisfying XH{t)x,t)H = xx,t)H, (1.41) are then easily seen to be related to the coordinate basis in the Schrodinger picture through x,t)H = e*tH x). (1.42) It is clear now that for t > t2 we can write H(xi,t1x2,t2)H = ( m l e - i ^ e t * 2 ^ ) = (xle-^-t ^H x2) = {xiU{h,t2)x2) = U(t1,x1;t2,x2). (1-43) This shows that the matrix elements of the time evolution opera- tor are nothing other than the time ordered transition amplitudes between the coordinate basis states in the Heisenberg picture. Finally, there is the interaction picture where both the quantum states as well as the operators carry partial time dependence. With- out going into any technical detail, let us simply note here that the interaction picture is quite useful in the study of nontrivially inter- acting theories. In any case, the goal of the study of quantum me- chanics in any of these pictures is to construct the matrix elements of the time evolution operator which as we have seen can be identified with transition amplitudes between the coordinate basis states in the Heisenberg picture.
- 29. 10 Field Theory: A Path Integral Approach 1.5 References Das, A., "Lectures on Quantum Mechanics", Hindustan Book Agency. Dirac, P. A. M., "Principles of Quantum Mechanics", Oxford Univ. Press. Schiff, L. I., "Quantum Mechanics", McGraw-Hill Publishing.
- 30. Chapter 2 Path Integrals and Quantum Mechanics 2.1 Basis States Before going into the derivation of the path integral representation for U(tf,xf,ti,Xi) or the transition amplitude, let us recapitulate some of the basic formulae of quantum mechanics. Consider, for sim- plicity, a one dimensional quantum mechanical system. The eigen- states of the coordinate operator, as we have seen in Eq. (1.36), satisfy Xx)=xx). (2.1) These eigenstates define an orthonormal basis. Namely, they satisfy (xx') = 5(x — x'), dxx)(x = l. (2.2) Similarly, the eigenstates of the momentum operator satisfying Pp)=pp), (2.3) also define an orthonormal basis. Namely, the momentum eigenstates satisfy (pp') = 8(p-p>), Jdpp){p = l. (2.4) l l
- 31. 12 Field Theory: A Path Integral Approach The inner product of the coordinate and the momentum basis states gives the matrix elements of the transformation operator between the two basis. In fact, one can readily determine that (px) = ^ e " ^ = (xW . (2.5) These are the defining relations for Fourier transforms. Namely, us- ing the completeness relations of the basis states, the Fourier trans- form of functions can be defined as f(x) = {xf) = Jdp{xp){pf) 1 [ V^ J ' dk el "x f{k), (2.6) f(k) = Vhf(p) f dxe~y;px f(x) -}=Jdxe-^f(x). (2.7) These simply take a function from a given space to its conjugate space or the dual space. Here k = | can be thought of as the wave number in the case of a quantum mechanical particle. (Some other authors may define Fourier transform with alternate normalizations. Here, the definition is symmetrical.) As we have seen in Eq. (1.42), the Heisenberg states are related to the Schrodinger states in a simple way. For the coordinate basis states, for example, we will have x,t)H = e%m x). It follows now that the coordinate basis states in the Heisenberg
- 32. Path Integrals and Quantum Mechanics 13 picture satisfy ^tHAtHJ H(x,tx',t)H = (xe-Kth e*tH x') = (xx') = 5(x - x1 ), (2.8) and ' tH dx x,t}H H(x,t = dxehtH x)( = e-nm fdxx){ xe h , xe~*tH = 1. (2.9) It is worth noting here that the orthonormality as well as the com- pleteness relations hold for the Heisenberg states only at equal times. 2.2 Operator Ordering In the Hamiltonian formalism, the transition from classical mechan- ics to quantum mechanics is achieved by promoting observables to operators which are not necessarily commuting. Consequently, the Hamiltonian of the classical system is supposed to go over to the quantum operator H(x,p)^H(xop,Pop). (2.10) This, however, does not specify what should be done when products of x and p (which are non-commuting as operators) are involved. For example, classically we know that xp — px. Therefore, the order of these terms does not matter in the classical Hamiltonian. Quantum mechanically, however, the order of the op- erators is quite crucial and a priori it is not clear what such a term
- 33. 14 Field Theory: A Path Integral Approach ought to correspond to in the quantum theory. This is the oper- ator ordering problem and, unfortunately, there is no well defined principle which specifies the order of operators in the passage from classical to quantum mechanics. There are, however, a few prescrip- tions which one uses conventionally. In normal ordering, one orders the products of x's and p's such that the momenta stand to the left of the coordinates. Thus, N.O. xp —> px , N.O. px —• px , 2 N.O. 2 x p —• px , N.O. o /r. 1 1 xpx —> px , (2.11) and so on. However, the prescription that is much more widely used and is much more satisfactory from various other points of view is the Weyl ordering. Here one symmetrizes the product of operators in all possible combinations with equal weight. Thus, xp —^ ~{xp + px), w.o. I , , . px —• -(xp + px), 2 W.O. I / 9 , , 2 x p —> -[x p + xpx + px ) , xpx —>' -(x2 p + xpx + px2 ), (2-12) and so on. For normal ordering, it is easy to see that for any quantum Hamil- tonian obtained from the classical Hamiltonian H(x,p) (x'HN -°-x) = f dp(x'p)(pHN -°-x) = J' ^.e-&*-^H{x,p). (2.13)
- 34. Path Integrals and Quantum Mechanics 15 Here we have used the completeness relations of the momentum basis states given in Eq. (2.4) as well as the defining relations in Eqs. (2.1), (2.3) and (2.5). (The matrix element of the quantum Hamiltonian is a classical function for which the ordering is irrelevant.) To understand Weyl ordering, on the other hand, let us note that the expansion of (axop + j3pop)N , generates the Weyl ordering of products of the form a^opPop naturally if we treat xop and pop as non-commuting operators. In fact, we can easily show that (*xop + (3pop)N = ] T - ^ an r(xn opP™)W -°- (2.14) n+m=N The expansion of the exponential operator e(aa;op+/3pop) would, of course, generate all such powers and by analyzing the ma- trix elements of this exponential operator, we will learn about the matrix elements of Weyl ordered Hamiltonians. Prom the fact that the commutator of a;op and pop is a constant, we obtain using the Baker-Campbell-Hausdorff formula Q 2 )ePPope( 2 I = gl 2 ) (APPop~ 2 ~ > — e(ax0p+/3p0p) _ (2.15) Using this relation, it can now be easily shown that dp (x e( 2 >epp °p p){peK 2 >x) / p-e-fc*-* C-^+0v). (2.16)
- 35. 16 Field Theory: A Path Integral Approach Once again, we have used here the completeness properties given in Eq. (2.4) as well as the defining relations in Eqs. (2.1), (2.3) and (2.5). It follows from this that for a Weyl ordered quantum Hamiltonian, we will have < z ' | # w - ° - ( W o p ) | , ) = f ^ e-i«°-*H {^f,p) • (2.17) As we see, the matrix elements of the Weyl ordered Hamiltonian leads to what is known as the mid-point prescription and this is what we will use in all of our discussions. We are now ready to calculate the transition amplitude. Let us recall that in the Heisenberg picture, for tf > U, we have U (tf,Xf]ti,Xi) = H(Xf,tfXi,ti)H. Let us divide the time interval between the initial and the final time into N equal segments of infinitesimal length e. Namely, let * = tj jr • (2.i8) In other words, for simplicity, we discretize the time interval and in the end, we are interested in taking the continuum limit e — > 0 and N — > oo such that Eq. (2.18) holds true. We can now label the intermediate times as, say, tn = U + ne, n = l , 2 , . . . , ( i V - l ) . (2.19) Introducing complete sets of coordinate basis states for every in- termediate time point (see Eq. (2.9)), we obtain U(tf,xf,ti,Xi) = H(xf,tfxi,ti)H = lim dx1---dxN-lH{xf,tfxN-i,tN-i)H N-^oo X H{XN-l-,tN-lxN-2,tN-2)H • • -H {xi,hXi,ti)H . (2.20)
- 36. Path Integrals and Quantum Mechanics 17 In writing this, we have clearly assumed an inherent time ordering from left to right. Let us also note here that while there are iV inner products in the above expression, there are only (N— 1) intermediate points of integration. Furthermore, we note that any intermediate inner product in Eq. (2.20) has the form H%nitnxn-i,tn—i)H = xne ft eft xn—) = { a g e - ^ - ' - ^ l x n - i ) = (xne~^eH xn-i) = f d Pn JpnjXn-Xn^-^H^-^^^) ^ ^ ^ Here we have used the mid-point prescription of Eq. (2.17) corre- sponding to Weyl ordering. Substituting this form of the inner product into the transition amplitude, we obtain U(tf,xf,ti,Xi)= lim / dxi • • • dajjv-i e->0 J dpi dpN 2-Kh'" 2nh N—>oo x e i E ^ K . » - 0 - « f i ( = ^ * . ) ) _ (2.22) In writing this, we have identified XQ = Xi, XN = Xf. (2.23) This is the crudest form of Feynman's path integral and is defined in the phase space of the system. It is worth emphasizing here that the number of intermediate coordinate integrations differs from the number of momentum integrations and has profound consequences in the study of the symmetry properties of the transition amplitudes. Note that in the continuum limit, namely, for e — > 0, we can write
- 37. 18 Field Theory: A Path Integral Approach the phase factor of Eq. (2.22) as N I™ ^ Yl [P^(x n ~x n-i) -eH ( ^n ' 1 , p n ) ) N^oo " n=1 N / / i. ^ ^ v / / Xn Xn— 1 TT I %n ~r Xn— hm - e £ [Pn [ J - H ( ,Pr = -*£dtip± =-[ H(x,p)) dt L. (2.24) U Namely, it is proportional to the action in the mixed variables. To obtain the more familiar form of the path integral involving the Lagrangian in the configuration space, let us specialize to the class of Hamiltonians which are quadratic in the momentum variables. Namely, let us choose H(x,p) = ^ + V(x). (2.25) n such a case, we have from Eq. (2.22) U(tf,xf,ti,Xi) = hm / dzi • • • dxN-i„ fc • N—>oo x e V v dpN 2-KK y/xn+xn. *)). (2.26) The momentum integrals are Gaussian and, therefore, can be done readily. We note that / A„ icfPn Pn<.x„-xn_1) U Pn t I 2m £ 2TT^ 6 / An i£ ( „1 2m Pn(xn-xn-l) a Pn e-2^h{Pn i 2irh
- 38. Path Integrals and Quantum Mechanics 19 " / 2nE e m(xn-xn_i)ji ( m(xn-xn_i) ' '-)-(•- 1 (2'Kmh2 gif1 ""1 "-1 = ( — Y 2mheJ (2.27) Substituting this back into the transition amplitude in Eq. (2.26), we obtain N_ U(tf,xf;tl,xi) = lim ( ^ ) 2 / iey^AT I m fx n-x n-l y(Xn+x n-l dxi • • • dxN-ieh n _ 1 V 2 ^ e ' ^ = A f Vx etti<l *m±2 -V ^ = AJvxe*s W, (2.28) where A is a constant independent of the dynamics of the system and S[x] is the action for the system given in Eq. (1.10). This is Feynman's path integral for the transition amplitude in quantum mechanics.
- 39. 20 Field Theory: A Path Integral Approach To understand the meaning of this, let us try to understand the meaning of the path integral measure Vx. In this integration, the end points are held fixed and only the intermediate coordinates are integrated over the entire space. Any spatial configuration of the intermediate points, of course, gives rise to a trajectory between the initial and the final points. Thus, integrating over all such configu- rations (that is precisely what the integrations over the intermediate points are supposed to do) is equivalent to summing over all the paths connecting the initial and the final points. Therefore, Feynman's path integral simply says that the transition amplitude between an initial and a final state is the sum over all paths, connecting the two points, of the weight factor eft ^J. We know from the study of quantum mechanics that if a process can take place in several distinct ways, then the transition amplitude is the sum of the indi- vidual amplitudes corresponding to every possible way the process can take place. The sum over the paths is, therefore, quite expected. However, it is the weight factor e^s ^x ' that is quite crucial and un- expected. Classically, we know that it is the classical action that determines the classical dynamics. Quantum mechanically, however, what we see is that all the paths contribute to the transition ampli- tude. It is also worth pointing out here that even though we derived the path integral representation for the transition amplitude for a special class of Hamiltonians, the expression holds in general. For Hamiltonians which are not quadratic in the momenta, one should simply be careful in defining the path integral measure Vx. 2.3 The Classical Limit As we have seen in Eq. (2.28), the transition amplitude can be written as a sum over paths and for the case of a one dimensional Hamiltonian which is quadratic in the momentum, it is represented as U(tf,xf;ti,xi) = A f Vxei5 ^ (2.29) mfxn-xn-l2 y(x n+x n-l e^O " J JV—xx> / lim A , rl'ri . . . H T A r 1 P
- 40. Path Integrals and Quantum Mechanics 21 where AN = (^k)2 • Even though one can be more quantitative in the discussion of the behavior of the transition amplitude, let us try to be qualitative in the following. We note that for paths where Xn ^ Xn—X, the first term in the exponential would be quite large, particularly since e is infinitesimally small. Therefore, such paths will lead to a very large phase and consequently, the weight factor can easily be positive or negative. In other words, for every such xn, there would be a nearby xn differing only slightly which would have a cancelling effect. Thus, in the path integral, all such contributions will average out to zero. Let us, therefore, concentrate only on paths connecting the initial and the final points that differ from one another only slightly. For simplicity, we only look at continuous paths which are differentiable. (A more careful analysis shows that the paths which contribute non- trivially are the continuous paths which are not necessarily differen- tiable. But for simplicity of argument, we will ignore this technical point.) The question that we would like to understand is how among all the paths which can contribute to the transition amplitude, it is only the classical path that is singled out in the classical limit, namely, when h — > 0. We note here that the weight factor in the
- 41. 22 Field Theory: A Path Integral Approach path integral, namely, eft *-x is a phase multiplied by a large quan- tity when h — • 0. Mathematically, therefore, it is clear that the dominant contribution to the path integral would arise from paths near the one which extremizes the phase factor. In other words, only the trajectories close to the ones satisfying iM=0 ' < 2 -3 0 ) would contribute significantly to the transition amplitude in the clas- sical limit. But, from the principle of least action, we know that these are precisely the trajectories which a classical particle would follow, namely, the classical trajectories. Once again, we can see this more intuitively in the following way. Suppose, we are considering a path, say #3, which is quite far away from the classical trajectory. Then, because h is small, the phase along this trajectory will be quite large. For every such path, there will be a nearby path, infinitesimally close, say #2, where the action would differ by a small amount, but since it is multiplied by a large constant would produce a large phase. All such paths, clearly, will average out to zero in the sum. Near the clas- sical trajectory, however, the action is stationary. Consequently, if we choose a path infinitesimally close to the classical path, the action will not change. Therefore, all such paths will add up coherently and give the dominant contribution as h — » 0. It is in this way that the classical trajectory is singled out in the classical limit, not because it contributes the most, but rather because there are paths infinites- imally close to it which add coherently. One can, of course, make various estimates as to how far away a path can be from the classical trajectory before its contribution becomes unimportant. But let us not go into these details here. 2.4 Equivalence with the Schrodinger Equation At this point one may wonder about the Schrodinger equation in the path integral formalism. Namely, it is not clear how we can recover the time dependent Schrodinger equation (see Eq. (1.30)) from the
- 42. Path Integrals and Quantum Mechanics 23 path integral representation of the transition amplitude. Let us recall that the Schrodinger equation is a differential equation. Therefore, it determines infinitesimal changes in the wave function. Consequently, to derive the Schrodinger equation, we merely have to examine the infinitesimal form of the transition amplitude or the path integral. Prom the explicit form of the transition amplitude in Eq. (2.29), we obtain for infinitesimal e U(tf = e,xf,ti = 0,Xi) /m^Kl^'-vpP)). (M1) 2-Kihe) v ' We also know from Eq. (1.38) that the transition amplitude is the propagator which gives the propagation of the wave function in the following way, / oo dx' U(e,x;0,x')il>(x',0). (2.32) -oo Therefore, substituting the form of the transition amplitude namely, Eq. (2.31) into Eq. (2.32), we obtain 1>(x, e) = ( ^ | - ) * | ° ° Me&l*-*'^f V ^H{x 0). (2.33) Let us next change variables to r] = x' - x , (2.34) so that we can write rP(x, e) = ( ~ ) h J°° dV e S ^ - ^ ^ + l ) ] ^ , +v, 0). (2.35) It is obvious that because e is infinitesimal, if 7 7 is large, then the first term in the exponent would lead to rapid oscillations and all such contributions will average out to zero. The dominant contribution will, therefore, come from the region of integration 0 < H < ( ^ ) (2.36)
- 43. 24 Field Theory: A Path Integral Approach where the change in the first exponent is of the order of unity. Thus, we can Taylor expand the integrand and since we are interested in the infinitesimal behavior, we can keep terms consistently up to order e. Therefore, we obtain x U(x, 0) + rtfix, 0) + tl/'{x, 0) + 0{rf) = (^)T*i im 2 a Ihe 'I -oo il>{x,0) --V(x)1>(x,0) ,2 (2.37) + # ' ( * , 0) + |-<(z,0)+O(r73 ,e 2 ) The individual integrations can be easily done and the results are f°° , iEL„2 f2mhe* / dr? e^ne7 ! = J-oo V ™ / / oo drirje^2 = 0, (2.38) •oo /•°° , 9 inL„2 ihe f2mhe2 I dr? 7 7 ezfie'7 = — J-oo m m J Note that these integrals contain oscillatory integrands and the simplest way of evaluating them is through a regularization. For example, / OO /"OO dr? e ^ 7 ' = lim / dry e^zsi" .oo <5^0+ J_00 (, V = (™**) (2.39) -OO 1 . . 1 = lim <5->0+ and so on.
- 44. Path Integrals and Quantum Mechanics 25 Substituting these back into Eq. (2.37), we obtain - ( — J ^"(x,0) + O(e 2 ) =tf>(x, 0) + ^ V (x, 0) - jV (x) i, (x, 0) + O (e2 ) o r ^ ( x , £ ) - ^ ( x , 0 ) = - | ( - ^ ^ + y ( a ; ) ) v ( x , 0 ) + O(62 ). (2.40) In the limit e — • 0, therefore, we obtain the time dependent Schrodinger equation (Eq. (1.30)) The path integral representation, therefore, contains the Schrodinger equation and is equivalent to it. 2.5 Free Particle We recognize that the path integral is a functional integral. Namely, the integrand which is the phase factor is a functional of the trajectory between the initial and the final points. Since we do not have a feeling for such quantities, let us evaluate some of these inte- grals associated with simple systems. The free particle is probably the simplest of quantum mechanical systems. For a free particle in one dimension, the Lagrangian has the form L = - m i 2 . (2.41)
- 45. 26 Field Theory: A Path Integral Approach Therefore, from our definition of the transition amplitude in Eq. (2.28) or (2.29), we obtain U(tf,xf,ti,Xi) = lim I ——-) / dxi • • • d ^ - i e h ^n = 1 2 ^ £ ) e^o V 2mne I J JV—>oo lim ( ) 2 / dxi • • •dxN-ie^'^n =l( -Xn ~Xn e->o v 2-nine J J i ) 2 TV—>oo (2.42) Defining / m 2heJ we have ^ = U i d * « , (2-43) U(tf,xf;U,Xi)= hm ( _ ) ^ _ J —>oo /"dyi • • -dj/jv-i e*E^=i(i/n-vn-i)a . (2.44) iV->oo X This is a Gaussian integral which can be evaluated in many different ways. However, the simplest method probably is to work out a few lower order ones and derive a pattern. We note that [dyx e*[(w-i»)2 +(i«-yi)a ] = [dyi eiPfoi-^T^+^w-w)2 ] ™*etto-vo) ( 2 .4 5 )
- 46. Path Integrals and Quantum Mechanics 27 If we had two intermediate integrations, then we will have / dyidy2 ei [^1 ~yo ^2+ ^2 ~?/1 ^+ ^3 '"?/2 ^2 ] = (*I* fdy2eih(y2-y0)2 +(y3-y2?] ) ' / d y 2 e ifte-^)2 +t(«)2 2 / V 3 ' (wr i , 2 2 ef(»/3-!/o) (2.46) A pattern is now obvious and using this we can write U(tf,xf,ti,Xi) e-+o 2irihe) m N e JV—>oo h m f^-)f f^V _l e-^o 2mne/ m I fl e N—*oo e2HNc (xN—xo) = h m ( ) e^hNeix f x i) e^o 2mhNeJ AT—>oo m 2irih(tf — ti) I i m (x f-x i) a (*/-*0 (2.47) Thus, we see that for a free particle, the transition amplitude can be explicitly evaluated. It has the right behavior in the sense that, we see as tf —> U, U{tf,xf,ti,Xi)->8(xf-Xi), (2.48)
- 47. 28 Field Theory: A Path Integral Approach which is nothing other than the orthonormality relation for the states in the Heisenberg picture given in Eq. (2.8). Second, all the potentially dangerous singular terms involving e have disappeared. Furthermore this is exactly what one would obtain by solving the Schrodinger equation. It expresses the well known fact that even a well localized wave packet spreads with time. That is, even the simplest of equations has only dispersive solutions. Let us note here that since S[x] = dt -mi;2 , J ti the Euler-Lagrange equations give (see Eq. (1.28)) This gives as solutions Xcl(t) = v = constant. (2.50) Thus, for the classical trajectory, we have S[xd] = f f dt ^mx = ^mv2 (tf - U). (2.51) On the other hand, since v is a constant, we can write Xf - Xi = V(tf - ti) or, v=X f^. (2.52) Substituting this back into Eq. (2.51), we obtain *M=Mt^)'<''-«-=^. (2 -53) We recognize, therefore, that we can also write the quantum transi- tion amplitude, in this case, simply as
- 48. Path Integrals and Quantum Mechanics 29 This is a particular characteristic of some quantum systems which can be exactly solved. Namely, for these systems, the transition amplitude can be written in the form U(tf,xf;ti,xi) = Ae^x ^, (2.55) where A is a constant. Finally, let us note from the explicit form of the transition ampli- tude in Eq. (2.47) that dU _ U im /xf — Xi ~dff ~ ~2{tf -U) ~~2htf-Uj dU _ im fxf — %i TT 2 / „ 2 dxf h tf — t d2 U _ im U (im fxf — Xi . dxi h tf — ti h J tf — t - J2 ^ ( h U 4- — fXf ~Xi 2 Tf] - V in 2{tf-U)+ 2 t f - u ) ) 2m {.BU . „ „ . Therefore, it follows that ..8U ti2 d2 U , „ „ . xh wr-^!*f> ( 2 -5 7 ) which is equivalent to saying that the transition amplitude obtained from Feynman's path integral, indeed, solves the Schrodinger equa- tion for a free particle (compare with Eq. (1.35)).
- 49. 30 Field Theory: A Path Integral Approach 2.6 References Das, A., "Lectures on Quantum Mechanics", Hindustan Book Agency. Feynman, R. P. and A. R. Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill Publishing. Sakita, B., "Quantum Theory of Many Variable Systems and Fields", World Scientific Publishing. Schulman, L. S., "Techniques and Applications of Path Integra- tion" , John Wiley Publishing.
- 50. Discovering Diverse Content Through Random Scribd Documents
- 51. an' we didn't git chucked off at a little side deepoe, same's the rest of the gang did." "Then the show is completely stranded?" "Jest that." "What's the name of the place?" "Ballardvale, I believe." "Hotel there?" "Dunno. We didn't stop to see." "Well, that was a miserable trick for Haley to play, but I guess most of the managers of traveling companies play it sometimes. Why did you chaps come here?" "We knowed you'd be here." "What of that?" "Waal, we reckoned mebbe we'd be able to git up some kind of a three-cornered show an' keep from starvin'. That was aour scheme. I dunno haow it'll hit ye, Frank." "I have just given a show at the opera house here." "Yeou hev?" "Yes." "Whut kaind of a show?" Then Frank explained just what had happened and what he had done, while his two friends listened in open-mouthed astonishment and admiration. "Jest like ye, by gum!" shouted Ephraim. "Can't throw yeou down! Yeou alwus light on yeour feet!"
- 52. "Yaw," nodded Hans, "yer veet alvays lighd on you, Vrankie." "Haow much money did ye make?" whispered Ephraim, eagerly. Frank pulled out a large roll, on the outside of which was a fifty- dollar bill. Both lads stared at it, and then they leaned heavily against each other. "Efy," whispered Hans, "I pelief I vos goin' to had a pad case uf heardt vailures!" "Waal, I'm ruther dizzy myself!" gurgled the Vermonter. "Never saw so much money as that in all my life. Why don't yeou retire an' live on the intrust of it, Frank?" "Yaw, why you don'd led der interest uf id life on you, Vrankie?" asked Hans. "Here is just about enough to get us started on the road in good shape," said Merriwell. "We shall need every dollar of it." "We!" squawked Ephraim. "Us!" gasped Hans. Merriwell nodded. "We will go into partnership," he said. "It will take three of us to run the thing right." The Yankee youth and the Dutch lad fell into each other's arms. "Saved!" cried Ephraim. "Dot's vot's der madder!" rejoiced Hans. "Oh, dot Vrank Merriwell vas a beach, you pet!" They sat down and talked it over for a long time. Frank believed Ephraim could learn to assist him about his tricks, and he fancied Hans would be good for something. They were his old Fardale
- 53. schoolmates, and he had no thought of leaving them stranded away out there so far from their homes. By the time they had talked over their plans it was after midnight. Then Frank found himself unable to deposit his money in the safe, as the clerk had gone to bed and taken the key, and no one would assume the responsibility of awakening him. Ephraim and Hans were given a room together. As they went upstairs, the Vermonter said to Frank: "Look aout for that air money, Frank. If yeou lose that, we're in the soup fer sure." "Oh, I'll look out for it," assured Merry. "No one will think of molesting me to-night." He little knew that these words were overheard by his worst enemy. From his own unlighted room Sport Harris peered forth, having the door slightly ajar. "So he's taking the money to his room?" thought the young scoundrel. "Well, he must have a pretty good pile of it, for that was a great house. I'm rather hard up, and I wouldn't mind lifting a fat roll off that fellow." In his stocking feet he slipped out into the hall and followed Frank, locating Merry's room. Frank went in, closed the door and locked it. He was pretty tired, and he lost little time in undressing. He did not give Sport Harris a single thought. In a short time he was in bed and the light was extinguished. Tired though he was, it was some time before Frank could get to sleep, for his brain was teeming with exciting thoughts. At last, however, he dropped off.
- 54. Frank awoke with a consciousness of danger. It seemed that a slight rustling had aroused him. In a twinkling he was on the alert, although he kept perfectly still. There was a sliding sound near the door. Turning his eyes, he saw a dark figure slowly slipping in through the transom, which was wide open. "Hello!" thought Frank. "Somebody is after my boodle! Well, I'll give that chap a surprise." He reached up near the head of his bed and pushed the button there, distinctly hearing the bell ring down in the office. Again and again he pushed it, determined to arouse somebody if possible. The intruder dropped down from the transom, and Frank shot out of bed. A second later Merriwell and the burglar were locked in each other's grasp.
- 55. CHAPTER IX. HARRIS AGAIN VANISHES. The burglar uttered a gasp of astonishment as Merriwell precipitated himself on the fellow. "Got you!" half laughed Frank. "I don't know!" The other twisted about like an eel. "Hold still!" "Not much!" The voice was choked by the efforts of the unknown, but Frank believed he recognized it. "So it's you, Harris!" he said. "Up to your old tricks! You are just as much a sneak as ever!" "If I'd got in before you discovered me, you might have never called me that again!" panted Harris. "By that I suppose you were bent on murder. Well, that is no worse than your record." "Why don't you shout?" hissed Harris. "Why don't you arouse the hotel?" "It isn't necessary." "Why not?" "Did you hear the bell ring in the office?"
- 56. "Yes." "I pushed the button. Somebody is coming here even now. All I have to do is to hold onto you till they come." Harris snarled and gnashed his teeth, which he tried to fasten in the back of Frank's wrist. "Steady," said Merry. "It's no use. I've got you, and I'll hold you. I'll see that you go to prison for this." "Never!" "It's what you deserve, and you'll have to take your medicine at last." Then Merry found his enemy was feeling in his bosom. Frank tried to hold his hand, but Harris succeeded in getting out a knife. With this he struck back at Merry. "That will look all the worse for you when they come," said Merry, grimly. "You are putting yourself in a pretty bad place." "Oh, I could kill you!" panted Harris. "You ruined my college career!" "You are wrong." "It is true." "You ruined it yourself." "No; you did it." "I did nothing of the sort. I gave you several opportunities to brace up and become a man, but you have bad blood in you, and blood will tell. I never did anything against you that you did not force me to do." "Oh, you will say that, but I know better. But for you, I'd be in Yale now."
- 57. "Yale is better off without you." With a sudden twist, Harris broke Frank's hold. A cry of triumph escaped him. "Now you get it!" The knife was driven at Merriwell's throat. Frank's hand caught his wrist, and the blade was stopped just as the point touched Merry's neck. Frank gave a twisting wrench, and the bones in the wrist of the young rascal seemed to snap. A cry of pain was wrung from his lips, and the knife fell clanging to the floor. There was a sharp knock on the door. "Wait a minute," called Frank. "I'll let you in directly. Got my hands full now." "What's the matter in there? What's this mean? Stepladder against the door out here." "Caller used it to come in with," cried Frank. Just then he found an opportunity to break away a bit from Harris, and he gave the fellow a terrible swinging blow. Frank's fist struck Harris under the ear, and the fellow was stunned. "Just lay there a moment," murmured Merry, as he dropped the baffled rascal on the bed and turned to open the door. The night watchman came in. Harris tried to get up and dart out by the open door, but Merry caught him and flung him back on the bed. "Just help me take care of him, will you?" said Frank. "He is pretty ugly, and——"
- 58. Over the foot of the bed went Harris, out of the half-open window he dived. Frank leaped and clutched at his heels. Too late! "Gone!" gasped Merry. "Well, it's more than even money that he won't go very far," said the watchman. "I'll wager something he's broken his neck by the fall to the ground." They hurried out of the room and down the stairs, fully expecting to find Harris lying under the window. But when they reached the spot both were amazed to discover that the fellow was not there! Nor was he found at all, although a sharp search for him was made. He had escaped again. Zolverein's remains were shipped to the little Eastern town that he sometimes called home, there to be interred in the village cemetery. Frank took care that everything was properly attended to, as he felt it his duty and privilege. M. Mazarin remained bitter toward Merriwell, and he disappeared almost as mysteriously as had Sport Harris. Frank proceeded to fill Zolverein's engagements, taking Ephraim and Hans along with him. "We're running a show of our own, now," he said, laughingly, "and we are out for fun, fame and fortune."
- 59. CHAPTER X. IN THE POWER OF HIS ENEMY. One eventful day Frank came alone to the theater for the purpose of getting something out of one of his trunks. Entering by the stage door, he went up the stairs and onto the stage, which was dark, behind the drop curtain. He discovered a man lifting from the easel on which it had rested the large mirror which was used in the "Educated Fly" trick. "Drop that!" shouted Frank. "All right!" The man promptly dropped the mirror at Frank's cry, smashing it into a thousand pieces! "Scoundrel!" Frank was aroused. "Back!" The unknown caught up a heavy Indian club, one of a set used by Merry each night in his exhibition of fancy club swinging. The club was raised aloft. "Back, or I'll brain you!" "Drop that!" "On your head, if I do!" The fellow made a threatening swing with the club. Frank ducked, dodged aside, leaped forward, caught his arm, grappled with him.
- 60. Now they were face to face, so close together that Merry could distinguish the features of the prowler. "Sport Harris!" he shouted, astonished by the discovery. "Yes!" snarled the other, trying to wrench his hand free. "You here?" "You bet!" "What for?" "Business." "Deviltry, more likely! How did you get in here?" "No matter." "Well, you'll pay dearly for that mirror!" "You'll never make me pay for it, you can gamble on that!" Now Harris made a furious struggle to break away, but Frank forced him back against some scenery and pinned him there. "It's no use, you rascal!" came from Merry's lips. "You are caught this time, and you won't get away." "Don't be so sure," panted Frank's enemy. "I have given you the slip more than once, and now——" He uttered a strange cry, and, a moment later, Merriwell realized there was danger behind him; but he was prevented from turning, and, all at once, a pair of small, strong hands encircled his throat, the fingers crushing into the flesh. Frank was in a bad scrape, as he instantly understood. Harris was not alone, and his companion had caught Merry unawares. "Choke him! choke him!" hissed Sport, with a savage laugh of satisfaction. "Now we've got him!"
- 61. Frank twisted and squirmed. For some seconds a furious struggle took place on that stage, but Harris managed to keep Merriwell from breaking the choking grip of the unknown, and those small, strong hands were crushing the life and energy out of the young magician. "Oh, we've got you!" exulted Frank's old Yale enemy. "You can't do it, Merriwell! You came here just in time to run your head into this trap!" Frank could make no reply, for his tongue was protruding from his mouth. In his ears there was a roaring sound, and colored lights seemed bursting and changing before his eyes. Frank knew the venom of Harris—knew the fellow was a brute who would hesitate at nothing to satisfy his evil desire for revenge. Alone he could have handled the young ruffian easily, but the attack from behind conquered him. He wavered, swayed, and would have fallen. They dragged him to a chair. "Ropes!" cried Harris. "Bring them quick! We'll tie him." The other hustled away and quickly returned. Then the two tied the unfortunate magician to the chair. "Something for a gag," called Harris. The other looked about, but could not find anything that suited Sport. "Oh, never mind," said the fellow, as he took a huge clasp knife from his pocket and opened it. "If he hollers, I'll cut his throat!" This was spoken in a way that seemed to indicate the ruffian would actually do the deed without hesitation. Harris drew up another chair and sat down facing the captive.
- 62. Slowly Merry's strength returned. At last he was able to sit up without the support of the binding ropes. "Ha! ha!" laughed his bitter enemy. "How do you like it? I don't believe you fancy it much. I have you now." Frank made no reply, but he peered through the gloom at the figure of Sport's companion and assistant. There was something familiar about the slight, supple form, but it was not till the man turned so the light reached him differently that Merry recognized him. "M. Mazarin!" he gasped, incredulously. The little man nodded. "Yes," he said, coldly. "Are you surprised to see me?" "Rather." "I suppose you expected never to see me again. You thought I had gone to leave you forever. You thought I would give up everything and let you go about the country giving exhibitions with this apparatus that should have become mine at the death of Zolverein. You fancied I was a fool. You robbed me of what should have been mine, and I do not love you for it." "Very fortunately," said Sport Harris, in his sneering way, "we met, became acquainted, discovered our mutual hatred for you. We are here—here to get even." "Right," nodded the little man. "If I can't take Zolverein's place on the road, I swear you never shall!" "It is plain that you make a fine pair," said Frank, speaking huskily, for his throat still felt the effect of the terrible pressure it had received. "You will do well together. Harris should have been in jail long ago, and it is not improbable you'll both get there before a great while."
- 63. "We'll ruin you before we go!" grated M. Mazarin. "It will take you a long time to duplicate this apparatus. Some of it you'll never be able to duplicate." "Are you going to steal it?" "Oh, no." "What——" "We are not that foolish," said the little man. "You might recover it if we stole it." "But you are going to do something?" "That's easy guessing," sneered Harris. "What is it?" "I will soon show you," said Mazarin, with a cold little laugh. "But you must keep him still, Harris." "If he utters a chirp, I'll slit his windpipe," promised the young ruffian. Mazarin lighted a lamp, which he placed on a small table. Then he took a heavy hammer, and before Frank's eyes he smashed at a single blow a box that served to enable Merry to do one of his most difficult and interesting feats. "Now," said the malicious little man, "you know what I am going to do. I am here to destroy every bit of the apparatus you received from Zolverein. I can do it in twenty minutes."
- 64. CHAPTER XI. DEADLY PERIL. Frank squirmed, and Harris laughed. "That hits you hard," said the fellow. "We'll soon put you out of business as a professional magician." "You shall pay dearly for every bit of property you destroy!" vowed Frank. "That's all right. You'll not worry anybody by talking like that. You'll have to catch your hare, and we'll be far away from here to- morrow." "I was too easy with you in the past, Harris," said Frank. "I can see that now." "Oh, yes, you were easy with me!" snarled the fellow. "You didn't do a thing but disgrace me in college! You——" "I simply exposed your tricks when you were fleecing my friends by playing crooked at poker. You brought it on yourself." "It's a lie! I didn't play crooked. I——" "You acted as the decoy to draw them into the game, while Rolf Harlow robbed them with his slick tricks. You can't deny that. You deserved worse than you received." "That's what you think. Anyhow, I'll have my revenge now. Go ahead, Mazarin; smash up the stuff." "He may shout."
- 65. "If he does, it will be his last chirp, for I swear I'll use the knife on him!" Frank fully believed the fellow would do just as he threatened. Besides that, it was extremely doubtful if anyone could hear him in case he shouted, as the theater was a detached building, in which there were no offices or stores. So Merriwell was forced to sit there, bound and helpless, and witness the destruction of his property, the intricate and costly apparatus for performing his wonderful feats of magic. With savage frenzy the little man battered and hammered and smashed the apparatus which had cost many hundreds of dollars. He laughed while he was doing it. Harris lighted a cigarette and sat astride a chair near Frank, whom he continued to taunt. "This is the finish of the career of Merriwell, the wonderful magician," he sneered. "He'll never be heard of again. Smash the stuff, Mazarin, old man! That's the way to do it! How do you like it, Merriwell? Doesn't it make you feel real happy to see him break up the furniture? Ha! ha! ha!" Now, not a word came from Frank, but his jaws were set and his eyes gleaming. It was plain enough that he had vowed within his heart that some day he would square the account with his enemies. Piece after piece of the apparatus was destroyed by the vengeful little man, while Harris sat and smoked, puffing the vile-smelling stuff into the face of the helpless youth. Since starting out to fill Zolverein's engagements on the road, Frank had been remarkably successful, but he could not go on without the apparatus, and it would take a long time for him to replace the articles thus maliciously ruined. Some of them he knew he would never be able to replace.
- 66. With the wrecking of his property one of his dearest dreams vanished. He had thought it possible that he might make enough money during vacations to carry him through Yale, so he could complete his course in college, which he had been forced to leave because of financial losses. He knew this was purely a speculation, as it was not certain he would continue to do a good business, especially when he got off Zolverein's route; but that had been his dream, and now it was over. Surely fate was giving him some hard blows, but still he did not quail, and he was ready, like a man, to meet whatever came. He had tasted of the glamour of the footlights, and there was bitter with the sweet. He had learned that the life of the traveling showman is far from being as pleasant and easy as it seems. But Frank had not started out in the world looking for soft snaps. He was prepared to meet adversity when it came and not be crushed. He felt that the young man who is looking for a soft snap very seldom amounts to anything in the world, while the one who is ready to work and push and struggle and strive with all his strength, asking no favors of anybody, is the one who is pretty sure to succeed in the end. Whenever fate landed a knockout blow on Frank he refused to be knocked out, but invariably came up smiling at the call of "time." It was plain that his enemies believed they would floor him this time and leave him stranded. Harris was watching Frank's face by the light of the lamp. "Oh, this is better than a circus!" chuckled the fellow, evilly. "Every blow reaches you, and I am settling my score." "Instead of settling it," said Merry, grimly, "you are running up a big account that I shall call for you to settle in the future."
- 67. "You'll have a fine time collecting." "But I always collect once I start out to do so." "Bah! Your threats make me laugh!" "Because I was easy with you in the past, you fancy I may be if my chance comes in the future. You are wrong!" "All bluff!" "Time will show that I am not bluffing now. I have given you more chances than you deserved. I shall give you no more. When next my turn comes, I shall have no mercy." Somehow Harris shivered a bit despite himself, for he knew that Frank Merriwell was not given to idle words. True, Frank had been easy with his enemies at college, but he must have changed since leaving Yale and going out into the world to fight the great battle of life. He had seen that the world gave him no favors, and now it was likely he would retort in the same manner. "Perhaps I may have no mercy now," said Harris. "You are in my power, and I can do with you as I choose. I am a stranger in this town. No one knows I am here. What if you were found in this old building with your throat cut? How could the deed be traced to me? Better spare your threats, Merriwell, for if I really thought there was danger that you would bother me in the future, I swear I'd finish you here and now!" Mazarin had finished his work of destruction. All the costly apparatus was broken and ruined, and the little man was standing amid the shattered wreck, wringing his hands and sobbing like a child that is filled with remorse after shattering a toy in a fit of anger. "All done!" he moaned; "all done!" Harris looked around, annoyed.
- 68. "What's the matter with you?" he fiercely demanded. "What are you whimpering about?" "I have broken everything!" "Well, now is your time to laugh." "Now is my time to cry! All those things should have been mine." "But were not." "No one can ever replace them." "And that knocks out Mr. Frank Merriwell. Wasn't that what you were after?" "But to have to smash all those beautiful things! I have broken my own heart!" "You're a fool!" Harris turned from his repentant companion, his disgust and anger redoubled. Frank, for all of the bitter rage in his heart, could see that Mazarin was not entirely bad. The little man's conscience was troubling him now. "I hate fools!" grated Harris. "I hate sentiment! A man with sentiment is a fool! You're a fool, Merriwell; you always were sentimental." "As far as you are concerned," spoke the captive, "I shall put sentiment behind me in the future. I am satisfied that you are irreclaimably bad, and you have the best chance in the world of ending your career on the gallows." "I don't care what you think." "I didn't suppose you would care. You are too low and degraded to care. In the past I spared you when you should have been exposed
- 69. and punished. Why? Because I hoped you would reform. Now I know there is no chance of that. For your own sake I spared you in the past; in the future, if my turn comes, for the sake of those with whom you will mingle and injure and disgrace, I shall have no mercy." These words, for some reason, seemed to burn Harris like a hot iron. His eyes glowed evilly, and he quivered in every limb. He leaned toward Merriwell, panting: "Your turn will not come! I might have let you go, but now——" He glanced down at the knife in his hand. Frank watched him closely, feeling all at once that the desperate wretch had formed a murderous resolve. Harris was hesitating. It was plain he longed to strike, and still his blood was too cold to enable him to bring himself to that point without further provocation. So he began to lash himself into fury, raving at Merriwell, striking Frank with his open hand, and repeating over and over how much he hated him. So savage did he become that Mazarin stopped his sobbing and stared at him in wonder. "You ruined my college career!" panted Harris. "You made me an outcast! You are the cause of all of my ill-fortune! And now you threaten to drag me down still further. You never shall! I'll see to that now!" He clutched Frank's shoulder and lifted the knife!
- 70. CHAPTER XII. RASCALS FALL OUT. "Stop!" The word came from Mazarin's lips, and the little man's left hand shot out and caught Sport's wrist, checking the murderous stroke, if Harris really meant to deliver it. "Let go!" "No!" The murderous-minded young villain tried to wrench away. He met with a surprise. The small, soft hand held him fast, despite all his writhings. Harris had wondered that Mazarin so easily choked Merriwell into helplessness, but, after twisting and pulling a few seconds and failing to break away, he began to understand the astonishing strength of those small hands. "What's the matter with you?" he snarled. "Are you daffy?" "You are, or you would not try that trick," shot back the little man. "Do you think I'm going to stand here and see you do murder? I guess not!" "It's my business!" "And mine now." "How?"
- 71. "If you killed Merriwell, I should be an accomplice. I'm not taking such chances." "You're a fool!" "No! you are the fool. I helped you get in here that we might square our account with him, not that you might cut his throat. You have lost your head. Do you want to hang?" "Of course not, but——" "Then have a little sense. I didn't think you rattle-headed. We are even with Merriwell now." "No, I shall not be even with him till I have disgraced him as he disgraced me!" hissed Harris. "I have brooded over it for months. I have dreamed of it. Sometimes I have been unable to sleep nights from thinking about it. I have formed a thousand plans for getting even with the fellow, and now——" "Now you would make yourself a murderer. Well, you'll have to choose another time to do that job. I am satisfied, and from this day I shall have nothing more to do with you." "So you are going back on me?" "No; I am going to quit you, that's all, for I am satisfied that you will get us both into a bad scrape if I stick by you." "All right; you can quit. You are too soft for me, anyway." Harris tried to show his contempt for Mazarin in his manner as well as his voice, but the little man did not seem at all affected. "You are too hard for me," he said. "I believe I was foolish in having anything to do with you." "Let go my wrist!" "Drop that knife!"
- 72. They now stood looking straight into each other's eyes, and there was something commanding in the manner of the little man who had smashed Frank's apparatus and then wept like a child over the ruin he had wrought. After some seconds, Sport's fingers relaxed on the handle of the knife, which fell to the floor, striking point downward and standing quivering there. Mazarin stooped and caught up the knife, closing it and thrusting it into a pocket. "Give it back," commanded Harris. "After a while," was the quiet assurance. "Not now. I don't care to trust you with it till——" He did not finish, but his meaning was plain. He believed Harris treacherous, and he would not trust the fellow till he was sure there would be no opportunity to use the knife on Merriwell. But Sport's rage had cooled, and now he himself was sick at heart when he thought how near he had been to committing murder. Passion had robbed him of reason for a time, but now cowardice robbed him of his false nerve, and he was white and shaking. Frank had watched the struggle between the two men with interest and anxiety, for he realized that his life might depend on the outcome. He fully understood that Mazarin had not saved him out of pity for him, but because the little man was more level-headed than his accomplice, and not such a ruffian. No matter if Mazarin did hate Merry, he was not ready to stain his hands with blood in order to satisfy his desire to "get even." A student of human nature, Frank understood Harris very well, and he saw when the reaction came. He knew well enough that all
- 73. danger was past when he saw the former Yale man grow white and tremble all over. In the past Merry had sometimes experienced a thrill of sympathy for the young gambler, understanding how youths who are fairly started on the downward course almost always find it impossible to halt and turn back. One crooked act leads to another, and soon the descent becomes swift and sure, leading straight to the brink of the precipice of ruin, upon which not one man in a thousand has the strength to check his awful career, obtain a foothold and climb back to the path of honesty that leads to the plain of peace. Now it was plain that Harris had sunk so low that there was little hope for him. He was almost past redemption. Incapable of feeling gratitude, the fellow had never realized that Merry had shown him any kindness in not exposing him and bringing about his disgrace when his crookedness was first discovered at college. Knowing that he would never let up in the least on an enemy, Harris had believed Frank "soft" because of his generosity. The fellow's hatred had grown steadily with each and every failure to injure Merriwell, while his conscience had become so hardened that he was not troubled in the least by things which might have worried him once. As Harris swung the knife aloft, Frank had braced his feet, preparing to thrust himself over backward as the only means of escaping the blow. This, however, had not been necessary, for Mazarin had interfered. "Now," said the little man, seeming to assume command, "it's time for us to get out of here." "I guess that's right," came weakly from Harris. "Some one might come."
- 74. "By this time it's dark, and we can slip out by the stage door without attracting attention." "We mustn't be seen coming out." "It's well enough not to be seen, but it wouldn't make much difference if we were. The people who saw us might think we were members of Merriwell's show." "Merriwell's show!" cried Harris, forcing a laugh. "I rather think his show business is over. We have put an end to that." Then he turned on Frank, some of the color getting back into his face. "We've fixed you this time," the revengeful fellow sneered. "It's the first time I've ever been able to do you up in good shape. You always managed to squirm out of everything before, but all your squirming will do you no good now." Frank was silent, his eyes fixed on Harris' face, and the fellow felt the contempt of that look as keenly as it was possible for him to feel anything. "Don't look at me like that!" he snarled. Frank continued to look at him. Once more Harris seemed losing his head. "How I hate you, Merriwell!" he panted, bending toward Frank, while Mazarin watched him narrowly. "I never dreamed I could hate anyone as I hate you." Then, quick as a flash, he struck Frank a stinging blow with his open hand, nearly upsetting the youth, chair and all. "Oh there is some satisfaction in that!" he grated.
- 75. "A coward's satisfaction," said the steady voice of the helpless victim. "Only a wretched coward would strike a person bound and unable to resist!" "That's right!" Mazarin uttered the words, and they filled Harris with unspeakable fury. "Right!" he snarled. "What's the matter with you? You smashed his stuff when he was tied and unable to prevent it. Was that cowardly?" "Yes!" Sport literally gasped for breath. "Yes?" he echoed. "What do you mean?" "Just that," nodded Mazarin, gloomily. "I have played the coward here, as well as you. I know it now, but it is too late to undo anything I have done." "Well, you make me sick!" Harris sneered. "You are one of the kind that does a thing and then squeals. I'm glad we are going to quit, for I wouldn't dare trust you after this." "Nor I you," returned the little man. "You'd be sure to do something to get us both in a mess. Come, are you going to get out of here?" "Directly." "Now?" "Wait a little." "What for?" "I have a few more things to say to Merriwell." "You have said enough. Let him alone."
- 76. "Well, we must gag him, or he will set up a howling the moment we are gone." "Let him howl. We'll be outside of the building, and it is dark. We can get away. It's not likely he'll be heard for some time if he does howl, and——" Slam! Somewhere below in the building a door closed. Harris made a leap and caught Mazarin by the wrist. "Somebody coming!" he hissed. "Sure thing!" "We must skip!" "In a hurry." "Which way?" There were steps on the stairs leading to the stage. Then Frank shouted: "Help! help! This way! Look out for trouble! Hurry!" "Satan take him!" hissed Harris. "He has given the alarm!" Mazarin did not stop an instant, but darted away amid the scenery and disappeared from view in the darkness. "Hello, Frank!" came a voice from the stairs. "Is that yeou? What in thunder's the matter?" It was Ephraim Gallup! "Look out, Ephraim!" warned Merriwell. "Enemies here! Danger!" Tramp, tramp, the Vermonter's heavy feet sounded on the stairs.
- 77. Then there was a rush, and a dark form swept down upon him, struck him, knocked him rolling and bumping to the foot of the stairs. "Waal, darn—my—pun—ugh!—kins!" came from the Yankee youth in jolts and bursts. Over him went the dark figure, closely followed by another. "Hold on a minute," invited Ephraim. "Whut's your gol darn rush?" But they did not stop. The door near the foot of the stairs was torn open, and both figures shot out of the building. Gallup gathered himself up. "Back broke, leg broke, shoulder dislocated, jaw fractured, teeth knocked out, tongue bit off, and generally injured otherwise," he enumerated. "All done in a jiffy. Whatever hit me, anyhaow? Hey, Frank!" From above Merriwell answered, and again Ephraim started to mount the stairs. He reached the top, found his way to the stage, and discovered Merry tied to the chair. "Good-evening, Ephraim," said Frank, grimly. "You are a very welcome caller. I'm getting tired of sitting here." "Hey?" gasped the Vermonter. "Whut in thunder——" He stopped, his jaw snapping up and down, but not another sound issuing from his lips. He was utterly flabbergasted. "Just set me free," invited Frank. "I'll tell you all about it later. Mazarin was one, Harris was the other. You've heard me speak of Harris. They caught me here, smashed my stuff, got away. We must catch them." "Gol dinged if I don't think so!" shouted the Yankee, and, a moment later, he was working fiercely to set Merriwell at liberty. Finding he
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