# On the origin of fermionic field

## Main Question or Discussion Point

Hello.

Standard Model is based on the lagrangian of the dirac equation where some gauge group are applied. But, i am asking me, why a field is used before it was created. Properties of particle come from the application of Noether Théorem and then a quantization. I imagine the same process for the creation of the field. In my mind i expect that matter fields are conseved quantities in a bigger manifold and the quantization produce quantum field theory ?

My question is : do you know the origin of matter field ? Or a paper speaking about ?
I read Michele Maggiore book (A Modern introduction to QFT) and recent papers without finding any answer.

Thanks a lot

Clément.

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As I understand it, the fields of QFT are simply introduced by hand for convenience. They are postulate to start with and only continue to be used because they happen to work. QFT is more of a reverse engineering effort than a derivation from mathematical principles.

I find myself wondering if there would be any acceptance of a theory derived from math alone even if it reproduced the equations that are commonly used. Perhaps that wouldn't even be consider science anymore.

strangerep
Properties of particle come from the application of Noether Théorem and then a quantization.
Well, the types of elementary particles correspond to the unitary irreducible representations of the Poincare group. (Since you've read Maggiore, I hope you understand what that means.) One does not need Noether's theorem to obtain these representations.

In general, representations of the Poincare group involve the Lorentz group, which is noncompact. Hence, one finds that the unitary irreducible representations of the Lorentz group are necessarily infinite-dimensional, i.e., field representations. That's where the fields "come from".

Lagrangians for interacting theories are then built from these basic fields, under various constraints such as preserving (a generalized form of) Poincare invariance.

• kroni
I don't understand (or read) this step it the book, Maggiore go fast and my english is not very good. Thank for this explaination ! I will work on it !

Ok, well, i read again Maggiore and some papper about the infinite dimensionnal représentation of Lorentz Group.
A infinite dimensionnal représentation satisfying constraint of the Lie algebra is the space of scalar field. Ok, we have find a infinite dimensionnal représentation, but i don't understand why it's a reason for the existence of these fields ? Because i see Lorentz group just like a group of transformation wich must leave invariant all équation, a symetry group. It's the reason why i invocate Noether théorem to créate field.

I feel that i don't understand something.

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ChrisVer
Gold Member
, but i don't understand why it's a reason for the existence of these fields ?
And I don't understand your question, about "existence" of these fields...what do you mean by existence? The answer is in post #3

It sounds like the OP is looking for a reason why such fields should exist in the first place... where does the electron come from. Maybe he needs to be instructed (as do I) why fields are postulated to begin with and what principles guide the invention of a field. We keep asking why. How far back can we go?

strangerep
A infinite dimensional représentation satisfying constraint of the Lie algebra is the space of scalar field.
Saying it that way is not quite right. The space of a scalar field (i.e., spin-0) is one possible representation of the Poincare group. But there are other possible spins, e.g., half-integral spin (fermionic), spin-1, etc.

Ok, we have find a infinite dimensional représentation, but i don't understand why it's a reason for the existence of these fields?
Anything physical must obey a sensible transformation rule so that different observers can sensibly compare their measurements. To transform between the reference systems of the (inertial) observers, we use the Galieli group (in the nonrelativistic case) or the Poincare group (in the relativistic case). A long time ago, Wigner had the deep insight that this means all elementary features of the world should correspond to the features of unitary irreducible representations of the relevant symmetry group.

Now,... do you understand that spatial isotropy (hence the 3D rotation group), combined with the requirement that operators of the 3D rotation are represented as operators on Hilbert space, means that only integral and half-integral spins are possible for elementary quantum systems? In other words, integral/half-integral spin is a feature of the unitary representations of the rotation group. If you don't understand this simpler example thoroughly, you could perhaps review Ballentine ch7, especially section 7.1.

The point here is this: the presence of a symmetry group (expressing the idea that different inertial observers experience the same physical laws), combined with the necessity for quantum representation on a Hilbert space, places severe constraints on the possible properties of elementary physical features of the world. Rotational invariance forces integral or half-integral spin. Poincare invariance then also forces the boson/fermion statistics distinction (via the spin-statistics theorem).

Because i see Lorentz group just like a group of transformation which must leave invariant all équation, a symmetry group. It's the reason why i invocate Noether théorem to créate field.
Noether's theorem does not "create" a field, since one already has fields when writing down a Lagrangian. Noether's theorem just tells you which quantities are conserved, given a particular Lagrangian.

I feel that i don't understand something.
Perhaps, like ChrisVer, I don't really understand your question.

StrangeRep, you perfectly understand my question and you answer perfectly. I did'nt know about the necessity of an unitary irréductible représentation, so it explain why we are looking for it. I understand better the way to field représentation.
I am an computer vision PHD, we work a lot with SO(3) to model rotation and use Lie algébra, générator and exponential map only to move on the manifold. I am very far from quantum mecanic and certain concept are hard to understand for me. May be to abstract.

I will read Ballentine ASAP !