The Advantage Of Changing The Lorentz Transformation To One With Imaginary Values

 In previous posts (on special relativity) we saw the format of the basic Lorentz transformation, e.g.:

x' = (x - vt)/ Ö[1 - v²/c²] 

t' = (t - vx/c² )/ Ö[1 - v²/c²] 

For x' and t' to be real numbers we require:  v <  c   

Further, we expect all the laws of physics to be invariant under these transformations. Recall  in plane Euclidean geometry rotations of a coordinate system are represented by matrices: 

A = A  (Θ) =

(a11.......a12)
(a21......a22)  

where Θ is the angle by which the system rotates.  A matrix defined by the special Lorentz transformation *  can then be written:

(1/Ö [1 - v²/c²]                 - v / Ö[1 - v²/c²])

( - v / c² Ö] [1 - v²/c²]        1/ Ö[1 - v²/c²])  

But consider if instead of the 2D coordinates x an t and x' and t' we take: x and ict, and x' and ict' - where we have introduced the imaginary number: i = Ö- 1

The earlier transformations then assume the form:

x' = 1/ Ö[1 - v²/c²]   (x +   iv/c  ict)

ict'  =  1/ Ö[1 - v²/c²]   (- ivx/c  +  ict)

Yielding the new matrix:

(1/Ö [1 - v²/c²]                (iv/c)   1 / Ö[1 - v²/c²])

( - iv / c)   1 /Ö] [1 - v²/c²]          1/ Ö[1 - v²/c²])

The above can then be transformed into simpler matrices in terms of:

cos iΘ  =   cosh Θ   =   1/ Ö[1 - v²/c²

sin iΘ  =   -i sin h Θ   =   ( iv / c)  (1/ Ö[1 - v²/c² ) 

Leading to the more compressed trig matrix:

.  

(cos iΘ..... sin iΘ)

(-sin iΘ)...cos iΘ)

Which can be thought of as rotation through an imaginary angle of the (x, ict) coordinate system. This is often a  more helpful form in formulating physical laws. 

Specifically, this is incredibly useful in theoretical physics for two main reasons: turning oscillating integrals into convergent ones and unifying quantum mechanics with statistical mechanics.

Digging deeper: In real i.e. (Lorentzian) time, the action factor (exp  iS/h) generates rapidly oscillating waves that are notoriously difficult to integrate, making rigorous calculations almost impossible.

By substituting time (t) with imaginary time (ict)), the Lorentz metric:

ds² =  -  dt²  +    dx²  

becomes a purely Euclidean metric :

ds² =  -  dt ²  +    dx²

Thereby the oscillating factor (exp  iS/h) transforms into the exponentially decaying factor (exp  -iS/h)

 Which makes the mathematical functions well-behaved and convergent, allowing physicists to rigorously solve associated path integrals.

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* Connecting systems with a common x-axis with velocity v directed along the same axis, for simplicity