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Quantum Field TheoryM.Sc. 4th Semester
MPHYEC-1: Advanced Quantum Mechanics
Unit III (Part 1)
Dr. Ashok Kumar Jha
Assistant Professor
Department of Physics, Patna University
Mob:7903067108, Email: [email protected]
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Lecture Outline
First and Second Quantization
What is the need of Quantum Field Theory?
Lagrangian formulation of Field
Assignment
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First Quantization
2
ipq , 1PBpq, Canonical
Quantization
Procedure
Particle aspect
of the objectFirst Quantization
Wave aspect
of the object
Particle Mechanics
q and p are coordinate & its
conjugate momentum
Quantum Mechanics
q and p are operators in
Hilbert space
Dynamical variables become operators;
E, L, … take on only discrete values.
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Second Quantization
Wave functions become field operators.
Properties described by counting numbers of 1-particle states being occupied.
Processes described in terms of exchange of real or virtual particles.
Systems with variable numbers of particles ~ Second quantization
High energy scattering and decay processes.
Relativistic systems.
Many body systems (not necessarily relativistic)
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What is Field?
Field associated with each kind of particle in nature satisfies an assumed
wave equation and it has infinite number of degrees of freedom.
A wave field is specified by its displacement at all points of space and
the dependence of these displacements on time.
Classical analogue: Weighted vibrating string with infinite number of
beads along the string, we come to limit of a continuous string described
by a displacement field which varies continuously with x and t. tx,
Road to Quantization of a Classical Field
(1) Start with the field equation
(2) Seek a Lagrangian via Hamilton’s principle
(3) Find the canonical momentum from the Lagrangian
(4) Carry out quantization procedure
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Why Quantum Field Theory?
Quantum field theory is a theoretical framework that
combines classical field theory, special relativity, and quantum
mechanics but not general relativity's description of gravity.
QFT is developed to study the behavior of particles moving with
comparable to speed of light and the size is at atomic size.
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Quantum Field Theory 6
Quantization of e m field photon (quantum of the field)
Structure to quantize the e m field is applicable to all
elementary particles as quantum fields
Quantum Field Theory establishes
link between the two descriptions
Classical wave
fieldSecond Quantization Quantum of the
field (particle)
Light is described in terms of
(1) electromagnetic field (classical description)
(2) photon (quantum mechanical description)
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Klein-Gordon Equation
7
022
2
2
m
t 1 c
22 mpE
(using natural units )
022
2
2
m
t
)()()( xxx
021 )()( tt
Klein-Gordon equation for free particle is given by
Energy eigen values are
Note: System will have no ground state and becomes unstable. Hence wave function interpretation of
Klein-Gordon equation is not possible.
Now we want to interpret as a field, so Klein-Gordon eqn for this
field is
Consider an infinitesimal variation
Restriction: & system is localized in space
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Real Scalar Field
8 022
2
23
2
1
mt
xddt
t
t
02
1 2222
32
1
mt
xddt
t
t
02
1
4
xxd
t
t
,L
x,tx
22
2
1
m
xxx,L
02
1 2242
1
mxx
xd
t
t
Hamilton’s principle 2
1
0
t
t
Ldt
L
xxxx
From Klein-Gordon eqn
Using the relation
Integrating by parts
This gives with
or
In analogy with Hamilton’s principle
is called Lagrangian density
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Euler-Lagrange Equation
9
Hamilton’s principle with
This gives
using
Integrating by parts
x,LL0
2
1
4 t
t
xd L
02
1
4
t
tx
x
xd
LL
02
1
4
t
t
x
xxd
LL
L
This leads to Euler-Lagrange equation
xx
The momentum conjugate to the real scalar field is given by
0
LL
x
x
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Assignment
1. Why we study quantum field theory?
2. What do you understand by First Quantization and Second Quantization?
3. Describe Lagrangian formulation of field equation.
4. Derive Euler-Lagrange Equation for field.
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Thank You