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Quantum Field Theory

Field Equation

Quantum field theory can be developed by adopting the path integral formulism as mentioned earlier, or by combining the field equation with canonical quantization as shown in the next sub-topic. Anyway, the concept of Action principle can be generalized to derive the field equation with some modifications for the action. A similar procedure for minimizing the action can be used to derive the Euler-Lagrange equation, which is now called the field equation. Thus, the action of the field is now expressed in the form:

where is the Lagrangian density, is the th component of the field, denotes the space-time component,

and the integration is over a four-volume R. Similar to the case of particle Lagrangan in Eq.(1a) S has to be a scalar if is an invariant function of its arguments. The form of is further constrained by demanding that it is invariant under certain transformations such as space-time translation, Lorentz transformation, the gauge transformation, or the conformal transformation, etc.... This is to ensure that the equation of motion (field equation) is unchanged (symmetrical) under these operations. The corresponding conservation laws for these symmetries are summarized in Table 01 below.

Symmetry Group Conservation Law
Space-time translation (Translational) Poincare (x x + a) Energy-momentum
Space-time rotation (Spatial) Lorentz Angular momentum
Interal rotation of complex field U(1) in scalar field Particle number
Internal rotation of lepton or quark field SU(2) in weak interaction 3rd component of weak isospin
Internal rotation of quark field SU(3) in strong interaction Number of color charges

Table 01 Conservation Laws

Note: The gauge symmetries in Table 01 are global (meaning everywhere), local gauge symmetries (at specific point) leads to the interactions with gauge bosons. See the sub-topic on Noether's Theorem for more details about charge conservation, and more about "Conservation Rules".

The "Action Principle" for field theories states that if we perform an arbitrary variation of the field, + , subject to the boundary condition = 0 for t = t1 and t = t2, then the solution of the variational problem (S) = 0 yields a set of the Euler-Lagrange equations from which the equations of motion, or field equations can be derived:
---------- (1k)
For the neutral scalar meson field

which yields the Klein-Gordon Equation according to Eq.(1k):
---------- (1l)
where m represents the mass of the one component scalar field , which is a function of x, y, z, and t (collectively represented by x in the equation),

are the d'Alembertian operator in 4-dimensional space-time and the Laplacian operator in 3-dimensional space respectively. The repeated dummy index in the equations is understood to be summed over the 4 space-time coordinates.

The field can be expressed in a series expansion in terms of the harmonic functions and the coefficients ck's, where k is a four dimensional vector related to the momentum and energy of the particle:
---------- (2)
which is just a Fourier Series where the coefficients are to be determined by the field:
---------- (3)
where , and the time x0 is set to zero after the time derivation has been performed.

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