Parametric Integration is an Integration technique that was popularized by Richard Feynman but was known since Leibinz’s times. But this technique rarely gets discussed beyond a niche set of problems mostly in graduate school in the context of Contour Integration.

A while ago, having become obsessed with this technique I wrote this note on applying it to Laplace transform problems and it is now public for everyone to take a look.

In a previous post, we took a look at Lorentz Invariant scalar fields by using Temperature as an example and derived some Lorentz invariant quantities. In this post we will do the same thing but by being not so rigorous.

where and are small infinitesimal change in coordinates.

But by virtue of series expansion we can express the second expression in the following way and find another Lorentz invariant quantity for scalar fields.

By a similar analysis, we can obtain

Using the above and the invariant distance ( ) we find that is also invariant under the Lorentz Transformation.

But what does this quantity physically represent and why is it important in physics ? That’s the question we will address in the next post of this mini-series.

In quantum mechanics you can denote the wave-function in the position or the momentum basis. Written in the momentum basis, it would look something like:

But momentum is a continuous variable and it varies from to .

Therefore changing to the integral representation we get that:

But is just the projection of the momentum vector on the wavefunction:

We are also aware from our knowledge of Fourier Transform* that the wave function written in momentum space is given as :

Comparing both the above equations if we take the momentum basis as , then:

We can perform a similar analysis by expanding the wavefunction about the position basis and get

** Where does the in the Fourier Transform come from ?

We know from Fourier transform is defined as follows:

Plugging in and rewriting the above equation we get,

We find that from

that the normalization constant is not but . Therefore,

Let’s consider a scalar field, say temperature of a rod varying with time i.e . (something like the following)

We will take this setup and put it on a really fast train moving at a constant velocity (also known as performing a ‘Lorentz boost’).

Now the temperature of the bar in this new frame of reference is given by where,

Temperature is a scalar field and therefore irrespective of which frame of reference you are on, the temperature at each point on the rod will remain the same on both the frames i.e

Therefore we can say that Temperature (a scalar field) is Lorentz invariant. Now what other quantities can we make from T that would also be Lorentz invariant ?

Is

Well, let’s give it a try:

——–

Clearly,

But just for fun let’s just square the terms and see if we can churn something out of that:

We immediately notice that:

Therefore in addition to realizing that is Lorentz invariant, we have also found another quantity that is also Lorentz invariant. This quantity is also written as .

Deeper meaning

We know that nature is relativistic and when we are are cooking up a Lagrangian for a theory, we better make sure that it is Lorentz invariant as well. What the above analysis on scalar fields hints us is that the Lagrangian for such a theory can contain terms like in it as the quantity does not change under a Lorentz transformation.

This discussion finds a deeper ground in Quantum Field Theory. For example if is a scalar field, then a Lorentz invariant Lagrangian could take any of the following possible forms:

All of them keep the action invariant under a Lorentz transformation.

The Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space and the trivial way to remember this is through an ansatz :

Let be a set of normalized basis vectors but we would also like to make them orthogonal. We will call be the orthonormalized set of basis vectors formed out .

Let’s start with the first vector:

Now we construct a second vector out of and :

But what must be true of is that and must be orthogonal i.e .

Therefore we get the following expression for ,

which upon normalization looks like so:

That might have seemed trivial geometrically, but this process can be generalized for any complete n-dimensional vector space. Let’s continue the Gram – Schmidt for the third vector by choosing of the following form and generalizing this process:

The values for and are found out to be as:

Therefore we get,

(or)

Generalizing, we obtain:

Now although you would never need to remember the above expression because you can derive it off the bat with the above procedure, it is essential to understand how it came out to be.

It is trivial for most astronomy textbooks to illustrate the parallax method as follows:

This is absolutely fascinating, but it was really hard to find actual images of stars in books that illustrate this.

This is the proper motion of 61-Cygni, a binary star system over a span of couple of years.

But Bessel discovered that in addition to this proper motion, 61-Cygni also wobbled a little bit from side to side because of the parallax during observation.

The following is a plot of the motion of 61 Cygni – A which beautifully elucidates the proper motion and the effect of parallax (i.e the wiggle of the blue line with respect to the mean free path)

In addition, if you would like to actually play around with data for yourself, the The Hipparcos Space Astrometry Mission might interest you a lot. The mission was Launched in August 1989 and successfully observed the celestial sphere for 3.5 years before operations ceased in March 1993 employ

The documentation and the catalogue are fairly clear , instructive and easy to use. Have fun!