Wolfram Alpha:

```World Lines Refresher
---------------------

Consider the LIGHT CONE in 2D.

Light-like (LL):  dτ2 = dt2 - dx2/c2 = 0  dt/dx = 1  ∴ v = c

Time-like (TL):   dτ2 = dt2 - dx2/c2 > 0  dt/dx > 1 ∴ v < c

= gμνdxμdxν

= g00(dx0)2 + g11(dx1)2 + g22(dx2)2
+ g33(dx3)2
-     -
| dt   |
Where gμν is of the form +--- and dxμ = | dx/c |
| dy/c |
| dz/c |
-     -

Space-like (SL):  dτ2 = dt2 - dx2/c2 < 0  dt/dx < 1 ∴ v > c

The space-like case is often written as:

ds2 = -c2dτ2

= -c2dt2 + dx2

= -gμνdxμdxν

-    -
| cdt |
Where dxμ = |  dx |
|  dy |
|  dz |
-    -

The proper distance, ds, is analogous to proper time. The
difference is that the proper distance is defined between
two spacelike-separated events (or along a spacelike path),
while the proper time is defined between two timelike
separated events (or along a timelike path).

Consider the world-line for a single point particle (no
PE terms).

Classical Action
----------------

A = ∫dt (1/2)m(dx/dt)2

Relativistic  Action
--------------------

World lines are always time-like (TL) curves in spacetime.
We would like to make the value of the action independent
of the parameter chosen to calculate it (the coordinate
time, t).  For this purpose we choose τ (the proper
time) since all Lorentz observers agree on its value.
This process is referred to as 'parameterization of the
world line'.  Starting with:

dτ = √(gμνdxμdxν)

We can write this as:

dτ = √(gμν(dxμ/dτ)(dxν/dτ))dt

Let's postulate that the action is:

A = -mc∫dτ

= -mc∫√(gμν(dxμ/dτ)(dxν/dτ))dτ

So L = (gμν(dxμ/dτ)(dxν/dτ))

Equations of Motion
-------------------

At this point we could apply the Euler-Lagrange equation
to derive the equations of motion.  If we did this we
would get the geodesic equation from GR:

d2x/dτ2 = Γσρμ(dxμ/dτ)(dxν/dτ) where Γ ~ ∂g

However, we can get at these equations in an easier way
by continung with our earlier postulate.

For flat space we get:

-          -  -   -
| 1          || cdt |
dτ2 = |   -1       ||  dx | = c2dt2 - dx2 - dy2 - dz2
|      -1    ||  dy |
|         -1 ||  dz |
-          -  -   -

Therefore, in one spatial dimension we get:

dτ = √(c2dt2 - dx2)

= √(c2dt2(1 - dx2/c2dt2)

= cdt√((1 - dx2/c2dt2)

= cdt√((1 - v2/c2)

Therefore, the action is:

A = -mc2∫dt √((1 - v2/c2)

Thus,

L = -mc2(1 - v2/2c2)

= -mc2 + (1/2)mv2 for v << c

For Rindler space we get:

-                 -
| (1 + 2U)  0  0  0 |
dτ2 = |    0     -1  0  0 |∂xμ∂xν
|    0      0 -1  0 |
|    0      0  0 -1 |
-                 -

Therefore,

dτ2 = (1 + 2U/c2)dt2 - dx2/c2

A = -mc2∫√((1 + 2U/c2)dt2/dt2 - (1/c2)dx2/dt2) dt
.
= -mc2∫√((1 + 2U/c2) - x2/c2) dt
.
= -mc2√(1 + (1/c2)(2U - x2))
.
= -mc2(1 + (1/2c2)(2U) - x2)
.
= -mc2 - mU + mx2/2

Where U = -MG/x

Quantum Mechanics
-----------------

For the classical point particle:

S = ∫ exp(i∫dt (∂x/∂t)2)

Path integrals in this form are referred to as functional.
Functionals are very difficult to solve but there is a
'trick' that can be used to help simplify matters.  The
'trick' is called ANALYTICAL CONTINUATION and involves
'continuing' the integral to an imaginary time variable
called a WICK ROTATION.  This makes the integral more like
that seen in statistical mechanics.  The Wick rotation
involves the substitution t = -it.

i∫dt (∂x/∂t)2 => i(-i)∫dt (1/-i)2(∂x/∂t)2

= -∫dt (∂x/∂t)2

So that S becomes,

S = ∫ exp(-∫dt (∂x/∂t)2)

The negative exponential now has the effect of suppressing
the contribution due to 'wild' trajectories.  This integral
is referred to as a PALEYâ€“WIENER integral with solutions of
the form F(s,s).```