# Lagrangian of a Photon: Understanding the Fundamental Particle in Light

• exmarine
In summary, photons have no Lagrangian, and so one cannot calculate anything about their action, amplitude, etc. without first calculating the action of the electromagnetic field and correlation functions.
exmarine
I can't find this in any textbook, so I must not understand something about it. What is the Lagrangian of a photon? Would it be just h*nu?

Photons have Spin 1. The general Lagrangian for Spin 1 particles is called the Proca Lagrangian and if put into the Euler Lagrange euquation yields the Proca equation. In addition, photons are massless. Therefore putting $m=0$ in the Proco yields the correct Lagrangian for photons. If you put this Lagrangian (i.e. the Proca with $m=0$ ) into the Euler Lagrange equation you get the inhomogeneous Maxwell equation.

You can find the Lagrangian, for example, here

There is no such thing as the lagrangian of a photon. Photons are quantum excitations of the electromagnetic field, which has a Lagrangian, essentially the lagrangian quoted by unknown1111.

bhobba
massless spin-1 = photon, carry only a kinitic term in L ;

L=-\frac{1}{4}F^{2}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=-\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}

OK, then how does one calculate the action (S) for the amplitude of a photon?

phi = (const) exp[(i/h_bar)S]

exmarine said:
OK, then how does one calculate the action (S) for the amplitude of a photon?
Which part of
Orodruin said:
There is no such thing as the lagrangian of a photon. Photons are quantum excitations of the electromagnetic field,
was unclear? You need to specify exactly what it is you are trying to do.

bhobba
Feynman & Hibbs, p. 29, eqn 2.15:

I can't seem to get the eqn editor to work.

Feynman & Hibbs, p.29, eqn 2.15: phi[x(t)] = const e^(I/h-bar)S[x(t)]

p.26: S = integral[L(x-dot,x,t) dt]

So if a photon has no Lagrangian, how does one calculate the action, amplitude, probability, etc. for a photon?

exmarine said:
So if a photon has no Lagrangian, how does one calculate the action, amplitude, probability, etc. for a photon?
You don't. You compute the action of the electromagnetic field and correlation functions (essentially amplitudes) between different excitations of the field.

exmarine said:
Feynman & Hibbs, p. 29, eqn 2.15:

Also, you are here assuming that we have the book available and ready to open. This is not getting us anywhere.

## 1. What is the Lagrangian of a photon?

The Lagrangian of a photon is a mathematical function that describes the energy and motion of a photon, which is the fundamental particle of light. It takes into account the photon's properties, such as its frequency and direction of travel, to determine its behavior.

## 2. How is the Lagrangian of a photon related to its properties?

The Lagrangian of a photon is directly related to the photon's properties. It takes into account the photon's energy, which is determined by its frequency, and its momentum, which is determined by its direction of travel. This allows us to understand how a photon behaves in different situations.

## 3. Why is understanding the Lagrangian of a photon important?

Understanding the Lagrangian of a photon is important because it allows us to make predictions about the behavior of light in various situations. This is crucial in fields such as optics, where precise control over light is necessary for technologies like lasers and fiber optics.

## 4. How is the Lagrangian of a photon derived?

The Lagrangian of a photon is derived using the principles of quantum mechanics and special relativity. It is based on the idea that the energy and momentum of a photon are related to its frequency and direction of travel, and that these properties are quantized, meaning they can only take on certain discrete values.

## 5. Can the Lagrangian of a photon be applied to other particles?

While the Lagrangian of a photon is specifically derived for the behavior of light, the same principles can be applied to other particles. In fact, the concept of a Lagrangian is used in many areas of physics to describe the behavior of particles, including electrons, protons, and even larger objects like planets and stars.

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