QUANTUM FIELD THEORY AND STANDARD MODEL
양자장론과 표준모형
Notes
LECTURE 1
What is Quantum Field and Why Quantum Field?
Classical Field Quantum Field
(wave in continuum limit)
Coherent motion of infinitely large number of degrees of freedom (thermodynamic limit)
Question : Why thermodynamic limit is important?
Example: Nearest neighbor Ising model
(Ref: N. Goldenfeld, Lectures on phase transition and the renormalization group
R. Shankar, Quantum field theory and condensed matter )
magnetic moment :
Since
No Magnetization ? : True for finite N
In one-dimensional case,
In thermodynamic limit, free energy density which is independent of N is well defined
Details:
Continuum limit : Heisenberg Model
Functional integral (a.k.a. path integral)
For particle physics,
Quantum Mechanics + Special Relativity
= Relativistic Quantum Mechanics
Newtonian Mechanics
Special Relativity
Non-relativistic
Quantum Mechanics
Relativistic
Quantum Mechanics
Why Relativistic QM is in the form of Quantum Field Theory (QFT) ?
Hint: Electrodynamics already contains special relativity
Example : Coulomb’s law + special relativity = Ampere’s law + Lorentz force
Elementary process :
Probability amplitude : S-matrix
Fock space appropriate for many-body physics
Question :
In order to describe the special relativistic S-matrix (or equivalently, probability amplitude), how the Hamiltonian should be given? That is, how the fundamental framework of relativistic quantum mechanics should be given?
Brief summary on Special Relativity
In two inertial frames, the interval between two events read
(Lorentz invariance)
“on (the mass-)shell”
1. Lorentz invariant integral measure
2. Fock space normalization
1) covariant normalization
2) non-covariant normalization
Translation is generated by the free Hamiltonian (caution!)
In covariant normalization
Time evolution
Interaction picture:
Time ordering :
Requirement :
1., 2. Probability amplitude is invariant under the action of Poincare group
Then we can impose
3. Causality of operator (NOT of state)
5. Cluster decomposition
: for far-separated processes, the probability amplitude factorizes
(see, e.g., Ch. 4, S. Weinberg, The Quantum Theory of Fields Vol. 1,
Ch. 6, A. Duncan, The Conceptual Framework of Quantum Field Theory)
We assume the process of stable spin-0 particle and
start with 5. Cluster decomposition
For this purpose, we define ‘connected interaction’
The Cluster decomposition imposes that
To be suppressed
For this requirement to be fulfilled, S-matrix must be in the form of
each term for the connected part must contain a single delta function only
: suppose S-matrix contains a term containing two delta functions then…
to see what the problem is, we replace the place wave with the wave packet
Indeed, interaction Hamiltonian contains at least single delta function (total energy-momentum conservation) due to the translation invariance
The cluster decomposition tells us that no more delta function appears.
This is guaranteed for local interaction :
Then how to construct the Hamiltonian consistent with the cluster decomposition?
: Introduce the creation / annihilation operator
Since creation/annihilation operator creates/annihilates a single particle, one can imagine the ‘particle field operator’ for a single particle which is linear in creation/annihilation operator.
Hamiltonian is regarded as a product of particle field operators in a local way :
Sum over all possible positions where interaction takes place
Hamiltonian density
Product of fields at the same position (local interaction)
Causality :
Consider
Condition 1 :
From
one finds
Condition 2 :
From (for non-covariant basis)
one finds
(for covariant basis, )
Condition 4 :
annihilation/creation of particle
annihilation/creation of antiparticle
(cf. some particles are their own antiparticles, e.g., neutral pion
while some neutral particles are not their own antiparticles, e.g., neutral Kaon )
SO(2) or U(1) symmetry: for real scalars
single complex scalar
Local quantum field for spin-0 particle :
annihilation operator for a particle
creation operator for an antiparticle
Plane wave as a solution to Klein-Gordon equation
(for higher spins, solution to equations of motion containing spin structure :
spin-1/2 : Dirac equation, spin-1 : Maxwell equation, spin-3/2 : Rarita-Schwinger equation
spin-2 :Einstein equation )