Elementary particles simultaneously exhibit properties of waves and
corpuscles. On one hand they diffract and interfere and on the other hand
they are always absorbed and generated as whole entities -- quanta. The
theory of elementary particles must therefore be a theory of quantum
fields.
Field
Field is a continuous distribution of a physical quantity in space and
time.
Field is a set φ of one or more (complex) numbers defined at every
point of space-time φ(t,r).
Example: electromagnetic field is
a set of four real numbers Aμ(t,r).
Classical Lagrangian field theory
Principle of least action
The physical field φ provides a minimum of the action integral,
S = ∫ L[φ,∂μφ]d4x ,
at which point the variation δS vanishes. Vanishing variation
leads to the Euler-Lagrange equations. The field φ that provides
a minimum of the action must satisfy the Euler-Lagrange equations:
∂μ [ ∂L/∂(∂μφ) ]
=∂L/∂φ.
Energy-momentum conservation
If the lagrangian does not depend explicitely upon time and coordinates
the energy-momentum conservation law follows,
∂μTμν=0,
where Tμν is the energy-momentum tensor:
Tμν=[∂L/∂(∂μφ)]∂νφ-Lδμν.
Principle of Relativity
All the laws of nature are identical in all inertial systems
of reference. Any law of nature has one and the same form in all
inertial systems. The equations expressing the laws of nature are
invariant with respect to transformations of coordinates and time from
one inertial system to another.
Building Invariant Equations for the fields requires knowledge
of Transformational Properties of the fields (from one inertial system
to another) which requires the knowledge of the Group of Coordinate
Transformations between inertial systems.
Exercises:
(Mulders, Ex.6.1a -- Klein-Gordon equation) Derive the
Euler-Lagrange equation for the real scalar field φ with the Lagrangian
L = ∂μφ∂μφ - m2φ2
(Mulders, Ex.6.2a -- Maxwell equations) Derive the Maxwell
equations from the Lagrangian