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For instance, how to find what internal symmetries a free relativistic lagrangian density has for complex scalar field?
ChrisVer said:You can find more symmetries. In case for the phi* phi, you can also have the parity transformation:
[itex] \phi \rightarrow - \phi [/itex]
This is one reason someone won't write [itex]\phi^3 [/itex] terms a priori in a Lagrangian. Or instead of a [itex]U(1)[/itex] you can have a broken [itex]U(1)[/itex] in the case that it would correspond to a [itex]Z_N[/itex]...
In fact you can find a lot of symmetries in a Lagrangian density. The reason is already stated, that the given Lagrangians will be built in order to contain your desired symmetries.
Take for example the quarks. The quarks are in a [itex]3[/itex]-dimensional representation of [itex]SU(3)_{color}[/itex]. How can you make neutral quantities from combinations of [itex]3[/itex]?
Take for example the [itex] 3 \otimes 3 = 6 \oplus \bar{3} [/itex]. Obviously the combination [itex]3 \otimes 3[/itex] cannot be decomposed to a singlet representation, but it contains the complex conjugate of [itex]3[/itex], that is the reason sometimes [itex]3 \otimes 3[/itex] are called anti-quarks.
On the other hand the combination [itex] 3 \otimes \bar{3}= 8 \oplus 1[/itex] contains the 1-dimensional object [which transforms trivially under [itex]SU(3)[/itex] transformations] and so can "work". The result is that terms which belong to [itex]3, \bar{3}[/itex] can be combined in the Lagrangian to give you allowed terms. Similarly for [itex] 3 \otimes 3 \otimes 3 [/itex] (so combinations of 3 fields belonging to [itex]3[/itex] representation).
ChrisVer said:What do you mean? a term like : [itex] \phi^2 \bar{\psi} \psi [/itex]?
one reason I see is that this term is non-renormalizable.
Internal symmetries are symmetries that exist within a system, such as a physical theory or mathematical equation, and do not depend on the system's external space or time. They are transformations that leave the equations or laws of the system unchanged.
Internal symmetries are important because they provide insights into the underlying structure and behavior of a system. They can simplify complex systems, help identify underlying principles, and lead to new discoveries and advancements in science.
Identifying internal symmetries can involve using mathematical techniques, such as group theory, to analyze the equations or laws of a system. This can also involve looking for patterns and similarities within the equations and identifying transformations that leave them unchanged.
Yes, internal symmetries can be broken. This can occur in certain systems when the symmetries are not preserved in all situations or at all energy levels. This breaking of symmetries can lead to a better understanding of the system and its behavior.
Internal symmetries are closely related to conservation laws. In fact, Noether's theorem states that every continuous symmetry in a system corresponds to a conservation law. This means that internal symmetries are fundamental in understanding the conservation of energy, momentum, and other important physical quantities.