Quantum Field Theory/Classical Lagrangian field theory

Special relativity

Special relativity was proposed by Albert Einstein in the beginning of the 20th century. The Special Theory of Relativity is a successor 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 $ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2}$ 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 $x^{\mu }$ itself is the energy-momentum or the momentum four vector $p^{\mu }=(E,\mathbf {p} )$. The dual of a fourvector $x^{\mu }$ is denoted by $x_{\mu }$. The dual vector $x_{\mu }$ is related to $x^{\mu }$ 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.

Variational principle

Action and Lagrangian

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

$S=\int L\,dt$

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:

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

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

Note that the variation $\delta \phi$ must not change the boundary condition of $\phi +\delta \phi$ 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).

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 $\phi$ 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.

Translation invariance, energy and momentum

Conserved current

• Canonical quantization of a scalar field
  • Lagrangian and Euler-Lagrange equation
  • Normal mode solutions
  • Generation and annihilation operators
  • Hamiltonian and commutation relations