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Quantum Field Theory

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Contents

[edit] Classical Lagrangian field theory

[edit] Special relativity

Special relativity was proposed by Albert Einstein in the beginning of the 20th century. The Special Theory of Relativity is a sucessor of Classical Mechanics, which is based on Newtonian mechanics, which was developed by Isaac Newton (as the name suggests). Classical mechanics is valid to a good accuracy in day-to-day phenomena involving speeds much less than the speed of light. However, at speeds comparable to the speed of light, classical mechanics breaks down. Classical mechanics is mainly based on invariance under Galilean transformations. This tells us how a phenomenon oberseved in one reference frame S would appear in another reference frame S^{\prime} which has a different velocity v relative to the original reference frame S. According to the Galilean transformation, the coordinates transform as follows:

x^{\prime}=x-vt

t^{\prime}=t

On the other hand, the special theory of relativity is based on invariance under the Lorentz transformation,

x^{\prime}=\gamma(x-vt)

t^{\prime}=\gamma(t-vx/c^{2}) where \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} Here, it is assumed that the reference frame S had a velocity with respect to S^{\prime} in x direction.

Note that under the Lorentz transformation, the interval ds2 = dt2 − dx2 − dy2 − dz2 remains unchanged. Or, in other words, the interval transforms like a scalar under the Lorentz transformation. The time and space coordinates together form a four vector x^{\mu}=(t,\mathbf{x}). Any quantity which transforms like the space-time coordinates under Lorentz transformation is defined as a four-vector. An example of a four-vector other than xμ itself is the energy-momentum or the momentum four vector p^{\mu}=(E,\mathbf{p}). The dual of a fourvector xμ is denoted by xμ. The dual vector xμ is related to xμ as x_{\mu}=(t,-\mathbf{x}). A product of a vector with a dual vector transforms like a scalar. Such a product is called as the inner product.

[edit] Variational principle

[edit] Action and Lagrangian

In classical mechanics, the action S and the Lagrangian L are related as follows:

S = \int L \, d t

These two quantities are defined similarly in quantum field theory. However, in quantum field theory it is often convenient to introduce a Lagrangian density \mathcal{L}. Hence the action can also be defined as:

S = \int\mathcal{L}\, d^4 x

[edit] Variational principle

One of the most important principles in physics which is also often called "Stationary Action Principle" or "Least Action Principle". Can be formulated in several ways:

  1. Of all possible fields with a given boundary condition the one that provides an extremum (often minimum, cf. Least Action) of the action is The Solution.
  2. The field for which the variation of the action vanishes is The Solution.

In other words if φ is The Solution and we add an arbitrary small variation δφ to it then the (linear part of the) variation of the action \delta S(\phi)\equiv S(\phi+\delta\phi)-S(\phi) vanishes, δS(φ) = 0.

Note that the variation δφ must not change the boundary condition of φ + δφ and must therefore vanish at the boundary.

Note also that the action must be real (just to talk about minima) and must be a 4-scalar (Lorentz invariant).

[edit] Euler-Lagrange equation

In classical mechanics, the Lagrangian L is a function of the canonical coordinates q and the canonical momenta \dot{q}. The Euler-Lagrange Equation is as follows:

\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}-\frac{\partial L}{\partial q} = 0

In quantum field theory, however, the two variables of the Lagrangian are the fields and the corresponding derivatives φ and \partial_{\mu}\phi. Furthermore, quantum field theory treats time and spatial derivatives at equal footing. Thus, the Euler-Lagrange Equation reads:

\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi\right)}\right) - \frac{\partial\mathcal{L}}{\partial\phi} = 0

where \mathcal{L} is the Lagrangian density.

[edit] Translation invariance, energy and momentum

[edit] Energy-momentum tensor

[edit] Conservation of energy and momentum

[edit] Hamiltonian

what s the hamiltonian

[edit] Conserved current

[edit] Transformational properties of fields

[edit] Relativity principle and the group of coordinate transformations. The group of Lorentz transformations.

[edit] Lie groups and Lie algebras. Lie algebra of the Rotation Group. Lie algebra of the Lorentz Group.

[edit] Group Representations. Irreducible representations of the rotation group. Direct product of two irreducible representations. Reduction of the direct product into a direct sum -- Clebsch-Gordan theorem.

[edit] Irreducible representations of the Lorentz group. Direct product of two irreducible representations.

[edit] Parity transformation. Bilinear forms of bispinors. Dirac matrices.

[edit] Quantization of free fields

[edit] Spin 0 field

[edit] Real and complex scalar fields. Klein-Gordon equation. Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Commutation relations.

[edit] Real and complex scalar fields.

The equations of motion for a real scalar field φ can be obtained from the following lagrangian densities

\begin{matrix} \mathcal{L}& = & \frac{1}{2}\partial_{\mu} \phi \partial^{\mu}\phi - \frac{1}{2} M^2 \phi^2\\ & = & -\frac{1}{2} \phi \left( \partial_{\mu} \partial^{\mu} + M^2 \right)\phi \end{matrix}

and the result is \left( \Box+M^2 \right)\phi(x)=0 .

The complex scalar field φ can be considered as a sum of two scalar fields: φ1 and φ2, \phi=\left(\phi_1+i\phi_2\right)/ \sqrt{2}

The Langrangian density of a complex scalar field is

\mathcal{L} = (\partial_{\mu} \phi)^+ \partial^{\mu}\phi - M^2 \phi^+ \phi

[edit] Klein-Gordon equation

Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above: \left( \Box+M^2 \right)\phi(x)=0

[edit] Spin 1/2 field

[edit] Dirac equation

The Dirac equation is given by:

\left(i\gamma^\mu\partial_\mu - m\right)\psi\left(x\right) = 0

where ψ is a four-dimensional Dirac spinor. The γ matrices obey the following anticommutation relation (known as the Dirac algebra):

\left\{\gamma^\mu,\gamma^\nu\right\}\equiv\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu = 2g^{\mu\nu}\times 1_{n\times n}

Notice that the Dirac algebra does not define a priori how many dimensions the matrices should be. For a four-dimensional Minkowski space, however, it turns out that the matrices have to be at least 4\times 4.

[edit] Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Anticommutation relations.

[edit] Spin 1 field

[edit] Massive spin 1 field. Additional (Lorentz) condition to eliminate spin-0.

[edit] Massless spin 1 field. Gauge invariance. Quantization within Coulomb (radiation) gauge.

[edit] Spin-statistics theorem. Discrete symmetries (C,P,T). CPT theorem.

[edit] Interacting quantum fields

[edit] Schwinger-Dyson equations

[edit] Interaction Lagrangians and coupling constans. Interaction representation for the time evolution of the state vector. S-matrix. Cross sections.

[edit] Covariant perturbation theory. T-product and N-product of the field operators. Propagators. Wick's theorem.

[edit] Interaction of fermions and bosons, Lint=-gψψ(φ+φ+). Feynman rules in coordinate and momentum space. Scattering of fermions. Non-relativistic limit. One boson exchange potential.

[edit] Introduction to The Standard Model

[edit] Local gauge symmetry. Gauge theories. Non-Abelian gauge theories.

[edit] Spontaneous symmetry breaking. Goldstone theorem. Higgs mechanism.

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