# Can particle physics be represented as an algebra?

Possibly the most useful thing anyone could tell me about particle physics:

Naively, one could try and make an algebra by enumerating all the types of particles and defining equivalence relationships (some rules), for example

Objects:

object type 1: photon

object type 2: electron

object type 3 ...

object type n

Rules:

rule 1: a photon is equivalent to an electron and a positron

rule 2: a proton is equivalent to three quarks

rule ....

rule n

Then one could study the properties of these relationships as a mathematical structure containing objects that represent particles and rules that define equivalence - i.e. as an algebra. I assume such an algebra would be, if formulated correctly, represented by Feynman diagrams (for some particles at least). Does such an algebra have a name? Is there more than one? Is this at all connected to the idea behind stating that the standard model is U(1) x SU(2) x SU(3)?

By an algebra I understand a closed and consistent collection of mathematical objects and relationships.

Thanks for trying to understand what I'm asking, with your input I can aim to make this more comprehensible. So I guess I'm looking for a formal structure which reflects particle reactions which is abstracted away from spatial, temporal and energetic considerations as far as is possible (is it at all possible?).

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I have trouble understanding what you're saying here. This site is geared for practical, answerable questions based on actual problems that you face. Combined with a typo in both the title and the standard model gauge group ⇒ -1. – Simon Nov 3 '11 at 0:45
If I understand correctly, the OP is trying to partition all known particles into two sets and then trying to establish some kind of (homo,iso) morphism between them based on known physics. I am not sure what the point of this exercise is? I don't think any question, however ill formulated deserves to be shot down. +1 – Antillar Maximus Nov 3 '11 at 2:09
@Antillar: Maybe I was a bit harsh... after the latest edit it does not annoy me so much! (Reversed my -1) – Simon Nov 3 '11 at 2:25
Lucas: What you're saying actually reminds me a bit of bootstrap models. But it is only indirectly related to the standard model gauge group. In that your equivalences would have to include the group representation aspects and more. – Simon Nov 3 '11 at 2:32
Simon: You got me looking at S-matrices, they seem pretty close to what I was thinking of, I need to look into it more to tell. – Lucas Nov 3 '11 at 2:44

What you want is an algebraic relation on particles which tells you which ones are allowed to turn into which other ones. These algebraic relations exist, and they are the conservation laws. The conservation laws come in two types--- discrete and continuous. The full list of exact conservation laws is known, and the exact ones are these:

• Energy/Momentum
• Angular momentum
• Electric charge
• CPT

In addition, there is one more conserved quantity, which is algebraically more complicated

• Strong color charge

This one is only applicable at short distances, and it tells you which collections of quarks and gluons can turn into which other collections of quarks and gluons. All objects are neutral at long distances.

In addition to these laws, there are two more nearly exact conservation laws:

• Generational lepton number (number of electrons+neutrinos of each generation)
• Baryon number (the number of end-stable protons plus neutrons)

There are further conservation laws which are not at all exact, but are preserved by the electromagnetic and strong interactions at low energies. These are parity, and charge conjugation.

All processes which obey the exact conservation laws are allowed in principle, although they might have vanishingly small probability amplitude.

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"These are parity, and charge conservation" I think you meant "charge conjugation". – mmc Nov 3 '11 at 3:32
@mmc: oops! Thanks. – Ron Maimon Nov 3 '11 at 3:33
So I would have a collection of objects defined by a set of quantum numbers and equivalences that are conservation laws. The non-exact ones are from vanishing small probabilities? – Lucas Nov 3 '11 at 3:49
@Lucas--- the "algebraic objects" you are talking about are the conserved quantities--- the sum of the charges for example. I don't know what you mean by "equivalences", you don't mean "equivalent", because a photon is not the same as an electron/positron. So you mean "can turn into". – Ron Maimon Nov 3 '11 at 4:18