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In Particle Physics, I've seen Scalar potentials which look like this $$ V = a \Phi^2 + b \Phi^4$$

$\Phi$ is scalar (a number).

What about vector potentials, and spinor potentials? How are they constructed? I know that when constructing expressions, a number of symmetries should be respected but I don't understand the whole story.

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Potentials are always scalars, so there is no such thing as a vector or spinor potential. There are things called the "vector potential", but this is something else. You can learn the representation theory of the rotation group as follows:

The expression you give from particle physics is not constructed by symmetry, it is constructed from renormalizability. This is the reason that it looks the way it does, with quadratic and quartic pieces only.

The quick and useful way to construct things that are invariant is to understand Einstein summation convention and tensors. If you want to make something that is invariant, it should follow the Einstein convention for index contraction.

There are quantities $T_{abcd...}$ which are tensors, you contract the indices to make the appropriate indices left over. The rotation invariance is SU(2), so that the tensor indices range from 1 to 2 and take complex values. Lorentz invariance is two SU(2)s, so there are two kinds of indices both going from 1 to 2, and the values are complex. In particle physics, you need to understand bigger groups, at the very least SU(3) for the strong interaction, but the index contractions are always how you produce invariants.

The other thing you need are the invariant tensors, which are those tensors whose components are invariant under the group. For SU(N) you have $\delta_{ij}$ and $\epsilon_{i....z}$ where the number of indices is equal to N. You can use invariant tensors in appropriate contractions to make invariant objects.

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@BB1: I see--- you are considering the "scalar potential" to be the contribution to the particle energy, and then its true that the vector potential adds to the momentum in the same way. I was considering the example of a Lagrangian term. I'll take away my downvote, but it needs an edit on your question. – Ron Maimon Jun 20 '12 at 16:55
This is my understanding of how vector potentials arise in QFT from studying David Tong's Lectures ( or), Susskind's Lectures on Youtube, and the first couple chapters of Zee's QFT in a Nutshell. I have no knowledge of vector potentials being involved in any other fashion. Therefore, I suspect the question is confused. But I admit that my knowledge is not as expansive as it could be. If my answer is wrong, then please explain why it is wrong because I'm curious. – MadScientist Jun 20 '12 at 17:52
@BB1: Your answer is ok, but it isn't usually the point of view people take regarding the vector potential. But it's a point of view I like very much! I only downvoted because I understood the question as being about group theory, not about vector potentials. The vector potential produces a potential momentum much like the scalar potential produces potential energy, and this is an important property, and one which comes across in your answer. I am sorry again for downvoting, I didn't understand what you were saying at first, and it looked completely unrelated to the question. – Ron Maimon Jun 20 '12 at 17:54
@ Ron Having reread your answer. I believe it is the same as mine, but more general. – MadScientist Jun 20 '12 at 18:09
@BB1: No it isn't. Your answer is still completely disconnected from the question, which is about group theory. It's just talking about the vector potential and how you introduce it in the point-particle action. This is a non-sequitor for this question, which is about how you make scalars out of vectors and tensors. – Ron Maimon Jun 20 '12 at 18:10

In Electrodynamics, the 4-Vector potential is the potential that gives the correct equations of motion via the Lagrangian. This calculation is done by applying Hamilton's Principle to the Lagrangian of Electrodynamics. So the problem can be rephrased as the question of what Lagrangian should be used and this is the Lagrangian that has the correct symmetries, is a relativistic invariant, and produces the correct equations (i.e. is verified by experiment). Because E&M is relativistic, the Lagrangian should involve a 4-Vector potential and this should be contracted (this is the relativistic method of obtaining scalars from 4-Vectors) in order to produce a scalar function to integrate over. The symmetries of the Lagrangian determine the conserved quantities via Noether's Theorem.

There are two standard procedures for moving from classical to quantum fields and these are the path integral approach and the method of second quantization.

The section on Electrodynamics in The quaternion group and modern P R Girard relates 4-Vector potentials to quaternions and spinors are related to quaternions so maybe this will be helpful.

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I wasn't asking about this, sorry maybe my question wasn't clear. But thanks anyways. – user09876 Jun 19 '12 at 18:43
@Ron I can't imagine a greater waste of time than googling and copy and pasting an answer to deceive strangers. – MadScientist Jun 20 '12 at 17:40
@BB1: Great! Unfortunately, other people do this. It's not intentional deception--- people want to help, they don't know the answer, they google, and think they find something relevant. That was my mistaken impression of what you were doing. My apologies. – Ron Maimon Jun 20 '12 at 17:46
That is understandable. For future reference, if I did that I would cite my source. I'm very careful about not plagiarizing. – MadScientist Jun 20 '12 at 18:06

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