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Classical Field Theory
----------------------
Massless Wave Equation
----------------------
A field in physics may be envisioned as if space
were filled with interconnected vibrating masses
and springs, and the strength of the field is like
the displacement of a ball from its rest position.
The theory requires "vibrations" in, or more
accurately changes in the strength of, such a field
to propagate as per the appropriate wave equation
for the particular field in question.
Consider an array of masses, m, separated by distance
ε and connected by massless springs each with
spring contant, k.
φ ^
| | o k_{i}
| ^ o o/\/\/o
| | | o
| o v o
|o | o o
| v
|________________________________________ x
<-ε->
Let ε be a small interval.
Let m_{i} = εm and k_{i} = K/ε. Thus, as ε -> 0 m_{i} -> 0
and k_{i} -> ∞.
Consider oscillations in the vertical direction.
.
KE, T = Σ_{i}(1/2)εmφ_{i}^{2}
PE, U = (K/2ε)Σ_{i}(φ_{i+1} - φ_{i})^{2}
Here it is assumed that the only contribution is
from neighbouring particles. Note that spring
force is inversely proportional to distance.
Lagrangian, L = T - V
.
= Σ_{i}(1/2)εmφ_{i}^{2} - (K/2ε)Σ_{i}(φ_{i+1} - φ_{i})^{2}
. .
∂L/∂φ_{i} = εmφ
The PE term requires a little more calculation:
U = (K/2ε)[(φ_{i+1} - φ_{i})^{2} + (φ_{i-1} - φ_{i})^{2}]
-∂U/∂φ_{i} = -(K/ε)[(φ_{i} - φ_{i+1}) + (φ_{i} - φ_{i-1})]
= (K/ε)[-φ_{i} + φ_{i+1} - φ_{i} + φ_{i-1}]
= (K/ε)[(φ_{i+1} - φ_{i}) - (φ_{i} - φ_{i-1})]
Apply the E-L equations to get the equations
of motion:
.
d/dt(∂L/∂φ_{i}) - ∂L/∂φ_{i} = 0
..
εmφ_{i} = (K/ε)[(φ_{i+1} - φ_{i}) - (φ_{i} - φ_{i-1})]
..
φ_{i} = (K/mε^{2})[(φ_{i+1} - φ_{i}) - (φ_{i} - φ_{i-1})]
In the limit ε -> 0, φ_{i} -> φ(x) and the RHS is
nothing more than the double derivative. Thus,
..
φ = (K/m)∂^{2}φ/∂x^{2}
Write v^{2} = K/m where v is the velocity along the
spring then,
∂^{2}φ/∂t^{2} - v^{2}∂^{2}φ/∂x^{2} = 0
This is the MASSLESS WAVE EQUATION.
Note the similarity between the wave equation
form and the invariant interval.
Lagrangians for Classical Fields
--------------------------------
A physical field can be thought of as the assignment
of a physical quantity at each point of space and time.
A field theory tends to be expressed mathematically
by using Lagrangians.
The Lagrangian for classical systems is a function of
generalized coordinates q and dq/dt. In this view,
time is an independent variable and q and dq/dt are
dependent variables in phase space. The action for
such a system is written as:
A = ∫dt L(q,dq/dt)
This formalism was generalized to handle field theory.
In field theory, the dependent variables q and dq/dt
are replaced by the value of a field at that point in
spacetime. Thus, q -> φ and dq/dt -> dφ/dt.
The action in this case is written as:
A_{φ} = ∫dtd^{3}x ℒ(φ,∂_{μ}φ) where ∂_{μ} = ∂/∂x^{μ}.
Where ℒ is the LAGRANGIAN DENSITY. Consequently,
L = ∫dx^{3} ℒ
Euler-Lagrange Equations for a Field
------------------------------------
Consider:
φ(x,t) = φ(x,t) + δφ(x,t)
Where,
δφ(x,t) = εf(x,t) and δφ_{μ} = ε∂f(x,t)/∂x^{μ}
Note: We have introduced the notation φ_{μ} ≡ ∂_{μ}φ
δA = ε∫d^{4}x[(∂ℒ/∂φ)f(x,t) + (∂ℒ/∂φ_{μ})(∂f(x,t)/∂x^{μ})] = 0
Integrate by parts to get:
δA = ε∫dtd^{3}x[∂ℒ/∂φ - ∂/∂x^{μ}(∂ℒ/∂φ_{μ})]f(x,t) = 0
Therefore, for this to be true for a non-zero
f(x,t), we must have:
∂/∂x^{μ}(∂ℒ/∂φ_{μ}) - ∂ℒ/∂φ = 0
This is the EULER-LAGRANGE equation for a field.
In 1 dimension this is:
∂/∂t(∂ℒ/(∂φ/∂t) - ∂/∂x(∂ℒ/(∂φ/∂x) ) - ∂ℒ/∂φ = 0
In general this is written as:
∂_{μ}(∂ℒ/∂(∂_{μ}φ)) - ∂ℒ/∂φ = 0
The momenta canonically conjugate to the coordinates
of the field (or simply the canonical momentum) is
defined as:
.
π = ∂ℒ/∂φ
Recall that the Lagrangian for the massless wave
equation is:
.
L = Σ_{i}(1/2)εmφ_{i}^{2} - (K/2ε)Σ_{i}(φ_{i+1} - φ_{i})^{2}
.
= ∫(1/2)φ^{2}dx - (K/2)Σ_{i}ε(φ_{i+1} - φ_{i})(φ_{i+1} - φ_{i})/ε^{2}
.
= ∫(1/2)φ^{2}dx - (K/2)Σ_{i}ε(∂φ/∂x)^{2}
.
= ∫(1/2)φ^{2}dx - (K/2)∫(∂φ/∂x)^{2}]dx
.
= ∫[(1/2)φ^{2}dx - (K/2)(∂φ/∂x)^{2}]dx
= ∫ℒdx
Where,
.
ℒ = (1/2)φ^{2} - (K/2)(∂φ/∂x)^{2}
ℒ is a scalar and is, therefore, Lorentz invariant.
Now ℒ may also contain additional terms to the
potential energy. For example, the system may be in
a gravitational field. This term can be defined as
follows:
V(φ) = (μ^{2}/2)φ^{2}
Therefore, ℒ becomes:
.
ℒ = (1/2)[φ^{2} - v^{2}(∂φ/∂x)^{2} - μ^{2}φ^{2}]
Or, in shorthand form:
ℒ = (1/2)[-∂_{μ}φ∂^{μ}φ - μ^{2}φ^{2}]
Where,
∂_{μ} = (∂_{t},∂_{x},∂_{y},∂_{z})
and,
∂^{μ} = (-∂_{t},∂_{x},∂_{y},∂_{z})
Apply E-L equations to get:
∂{∂((1/2)(∂φ/∂t)^{2}}/∂t = ∂^{2}φ/∂t^{2}
∂{∂((1/2)(∂φ/∂x)^{2}}/∂x = ∂^{2}φ/∂x^{2}
∂ℒ/∂φ = μ^{2}φ
Therefore,
∂^{2}φ/∂t^{2} - v^{2}∂^{2}φ/∂x^{2} + μ^{2}φ = 0
\ \
Field KE Field PE
Without the μ term this becomes:
∂^{2}φ/∂t^{2} - v^{2}∂^{2}φ/∂x^{2} = 0
So, not surprisingly, solutions to the E-L equation
for a simple field is the massless wave equation.
Solutions of this equation are of the form:
φ = f(x + vt) + g(x - vt) waves with right and left
moving parts. Note that wave shapes are preserved
as they pass through each other. If the m term is
added then the waves scatter each other.
To appreciate the content of this equation, consider
the connection with quantum mechanics:
p = -ih∂/∂x ∴ ∂^{2}φ/∂x^{2} = p^{2}/h^{2}
E = (ih)∂/∂t ∴ ∂^{2}φ/∂t^{2} = -E^{2}/h^{2}
We can write:
(E^{2} - c^{2}p^{2} - m^{2}c^{4})φ = 0
with m^{2}c^{4}/h^{2} = μ^{2}
This is the Klein-Gordon equation.