Title: The essence of Particle Physics
1The essence of Particle Physics
2The essence of Particle Physics
Particles are actually not like balls but
essentially more fields!
Well, not quite.
3They are quantized fields.
Fields when quantized are not like fields but
more like particles.
Quantum Field Theory
4Field Theory
Space and time are treated equally as parameters.
It is manifestly Lorentz Invariant.
Quantum Field Theory
For a quantum theory of field, the field is
promoted to operators!
It is still manifestly Lorentz Invariant.
Particle Quantum Mechanics
Space is operator while time remains a number
parameter.
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6Classical Field Theory
It is just like electric field but simpler, as a
scalar not a vector.
? is a scalar not a vector as electric field
It is easiest to describe fields using Lagrangian
and Hamiltonian.
Action is defined as the integral over time of
the Lagrangian.
The equation of motion is given by the principle
of Least Action.
For fields, the Lagrangian would be the integral
over space of a Lagrangian Density L
The integration is Lorenz Invariant.
The Lorenz invariance of the Lagrangian density
will guarantee the Lorenz invariance of Action
and hence EOM.
7The equation of motion is given by the principle
of Least Action.
Euler Equation
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Klein-Gordon Equation
8Hamiltonian Formalism
For Fields
Conjugate Momentum
9For Klein-Gordon Fields
10Solving KG Equation
Expand the KG field in terms of Fourier Series
Plug into KG Eq.
Every Fourier Component behaves like a SHO with ?
KG Field is just a collection of SHOs.
Each SHO is characterized by its k or p
momentum.
The frequency ? or energy of the SHO is just
that of a relativistic particle with mass m.
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13These SHOs correspond to the plane wave
solutions of KG Eq.
A general solution is a linear superposition of
all plane waves.
14The solution of KG Equation
For real field
The real solution of KG Equation
15Dirac Notation
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Canonical Commutation Relation
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25Now! Quantum Field Theory
26We use Canonical Quantization to go from
mechanics to quantum mechanics
Upgrade all observable to operators and impose a
commutation relation between position and
momentum
27Fields grow out of systems of particles
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Space coordinates x are actually indices!
We know how to quantize particle system and hence
we know how to quantize fields!
28Upgrade all observables to operators and impose a
commutation relation between fields and their
momenta
Quantum Field Theory is done!
29What is the commutation relation of the a
operators?
30KG Field is just a collection of SHOs.
Hint from Quantum SHO
31Reasonable Guess
SHO of different p are decoupled and hence their
operators commute.
32The operator a can be used to raise the energy
by one quantum while the operator a can be used
to lower the energy by one quantum
33The operator a is called Raising Operator while
the operator a Lowering Operator.
34(No Transcript)
35Quantum Field Theory is just a series of quantum
SHO.
The operator ap can be used to raise the energy
by one quantum ?p while the operator ap can be
used to lower the energy by ?p.
36There is a conserved momentum.
The operator ap can be used to raise the
momentum by one quantum p while the operator ap
can be used to lower the energy by p.
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38Particle space are built.
ap Creation operator and ap Annihilation
operator of a particle with momentum p and energy
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39Dirac field and Lagrangian
The Dirac wavefunction is actually a field,
though unobservable!
Dirac eq. can be derived from the following
Lagrangian.
40Negative energy!
41Anti-commutator!
A creation operator!
42b annihilate an antiparticle!
43Exclusion Principle