Deriving the equations of QM/QFT

[Tio Barnabe]

It seems that one can derive all the fundamental equations of QFT and, consequently, of non relativistic QM, by requiring $U(1), SU(1),SU(2),SU(3)$ symmetries. Also, it seems that one can succeed in doing so if one considers the irreducible representation of those groups.

My question is, Do we obtain incorrect equations if we don't use the irreducible representations? Unfortunately, I'm unable to try it myself.

 

[Demystifier]

First, SU(1) is a rather trivial group and it does not play an important role in physics.
Second, in your list of groups you are missing the group of spacetime symmetry, i.e. Lorentz or Poincare group.
Third, reducible representations are also used. For instance, Dirac spinors correspond to a reducible representation.
Fourth, physics cannot be derived only from symmetries. One also needs some physical principles, such as quantum theory per se.

 

[Tio Barnabe]

Second, in your list of groups you are missing the group of spacetime symmetry, i.e. Lorentz or Poincare group.

But is'nt it $SU(2) \times SU(2)$?

 

[Demystifier]

But is'nt it $SU(2) \times SU(2)$?

Mathematically yes (at least locally). But physically, from the fact that gauge group is $U(1)\times SU(2)\times SU(3)$ you cannot conclude that spacetime group is related to $SU(2) \times SU(2)$.

 

[Tio Barnabe]

you cannot conclude that spacetime group is related to $SU(2) \times SU(2)$.

So what group is it related to?

 

[vanhees71]

It's rather $\mathrm{SL}(2,\mathbb{C})$. It's called the covering group of the special orthochronous Lorentz group $\mathrm{SO}(1,3)$. Analogously the rotation group is $\mathrm{SO}(3)$ with the covering group $\mathrm{SU}(2)$.

BTW: U(1) is also not that unimportant. E.g., electromagnetism is based on the Abelian gauge group U(1)!

 

[Demystifier]

So what group is it related to?

You missed my point entirely. Gauge symmetry is physically unrelated to spacetime symmetry. Mathematics of groups is one thing, their physical realization is another. The rotations in space described by SO(3) are physically unrelated to gauge group SU(2), even though, mathematically, those two groups are locally the same. Physics is much more than representations of groups. See again my fourth point in post #2.

 

[PeterDonis]

U(1) is also not that unimportant

U(1) is important, yes. But SU(1) is trivial, as @Demystifier said. Note that the OP included both, which is not necessary; just including U(1) is enough.

 

[PeterDonis]

one can derive all the fundamental equations of QFT and, consequently, of non relativistic QM, by requiring $U(1), SU(1),SU(2),SU(3)$ symmetries

No, you can't. You don't derive the fundamental equations of QFT from symmetries. You have to derive them by other means, to obtain a framework within which you can write down specific Lagrangians based on particular symmetries. (This is another way of putting the fourth point @Demystifier made in post #2.)

 

[Tio Barnabe]

No, you can't. You don't derive the fundamental equations of QFT from symmetries

By fundamental equations I really mean the most important lagrangians, e.g. those describing interaction between 1/2 and 0 spin fields, 1/2 and 1, etc... i.e. dirac lagrangian, proca lagrangian, etc...

 

[PeterDonis]

By fundamental equations I really mean the most important lagrangians

Ok, but it should be understood that just knowing the Lagrangian doesn't help unless you know what to do with it. You need to derive the framework of QFT, that tells you what to do with the Lagrangian, by other means; symmetries won't help with that.

 

[Tio Barnabe]

You need to derive the framework of QFT, that tells you what to do with the Lagrangian, by other means; symmetries won't help with that.

Can you give a specific example?

 

[PeterDonis]

Can you give a specific example?

A specific example of what? If all you have is a Lagrangian, knowing nothing else about the equations of QFT, what would you do with it? I can't give you an example of what to do in that case because my answer to the question I just posed would be "Nothing, because I don't know what to do with a Lagrangian if that's all I have."

 

[Tio Barnabe]

what would you do with it?

I don't know what to do with a Lagrangian if that's all I have

You just use the Euler-Lagrange equations to get the equations of motion?

 

[PeterDonis]

You just use the Euler-Lagrange equations to get the equations of motion?

In classical physics, yes. Not in QFT.

Even in classical physics, what is it that tells you to use the Euler-Lagrange equations to get the equations of motion? It's not symmetries.

 

[Tio Barnabe]

Not in QFT

Are you sure? Please, have a look http://www.physics.purdue.edu/~clarkt/Courses/Physics662/ps/qftch21.pdf

 

[PeterDonis]

have a look

It says that classical field theory uses the Euler-Lagrange equations. It doesn't say that quantum field theory does.

 

[vanhees71]

Well, I'd say the Hamilton principle of least action (I'd rather call it stationary action), is just a method to formulate the dynamics in a broad range of physical systems. On a fundamental, to our present understanding, all of physics can be formulated in this way.

Further it turns out that the Hamiltonian formulation, i.e., dynamics in the (locally) symplectic phase space is the true mathematical framework in fundamental physics, and it's the most convenient starting point for the heuristics of quantum theory ("canonical quantization").

In other words, the Hamilton principle is a kind of ordering scheme of the possible dynamical equations to describe physical systems. Another important point are that it enables an elegant and natural scheme to describe symmetries. Since Einstein's famous first sentence in the famous relativity article of 1905, the analysis of symmetries of the natural laws in terms of Lie groups and Lie algebras (and, more recently, their generalization to $\mathbb{Z}_2$ graded algebras, aka "supersymmetry") has become of prime importance. Among the very rich possibilities to invent equations of motion derived from the variational principle those have been found most successful in describing the physical world which are constrained by symmetry principles.

All of theoretical physics, if presented a posteriori in a deductive way in a "quasi-axiomatic" formulation, starts with a (so far classical) spacetime model which implies some geometric structure and thus also symmetries. So for any physical theory, using a given spacetime model, they should obey the symmetries of this spacetime model in order to be consistent with it. Since Lie symmetries imply conservation laws (one of Noether's theorem) each one-parameter subgroup of the spacetime symmetry group implies a conservation law, and in Newtonian or special-relativistic space time this implies the 10 conservation laws (energy, momentum, angular momentum, and center-of-mass/momentum motion), valid for any closed system (in General Relativity, where Poincare symmetry is realized only locally, this holds only locally).

Last but not least, when making the heuristic step from classical to quantum theory, the symmetry principles are the only systematic way to (admittedly a posteriori) understand, why the quantum theories look as they do, since the operator algebras governing the specific structure of either non-relativistic QM or relativistic QFT. Canonical quantization has been the way to heuristically find the concrete theory of QM and relativistic theory, but the symmetry principles tell us, why these should be the inevitable consequences of "quantizing" a classical theory. In some sense they are the true "correspondence principles" making the heuristic way of "quantization" possible, and canonical quantization is also not a safe ground to study more complicated systems. Already the theory of quantizing the rigid body (solid top), leads to wrong equations when done naively without the guide from the rotation group, finally leading to applications in, e.g., analyzing the rotation spectra of molecules, etc.

As it turned out, also other even more abstract symmetry principles are of great importance in all physical theories, most importantly in elementary-particle physics, where the analysis of the empirically found conservation laws, concerning "intrinsic quantities" like electric charge, baryon and lepton number, etc. plays the role of an ordering principle, bringing order in the plethora of phenomena, particularly the "zoo of hadrons" found in various particle accelerators.

Of course, Hamilton's principle alone is a pretty general but empty scheme to formulate and then analyze dynamical laws, formulated in terms of differential equations, leading naturally to symmetry principles (thanks to Noether's work of 1918). Physics after all is an empirical science, and you need a lot of observations to figure out the right symmetry principles to finally formulate a theory like the Standard Model of elementary particle physics that can be taught within 1-2 semesters in our physics curricula, summarizing decade-long empirical and theoretical work!

 

[Demystifier]

You just use the Euler-Lagrange equations to get the equations of motion?

Why do you use the Euler-Lagrange equations? Euler-Lagrange equation is not a consequence of symmetry.

 

[vanhees71]

As I said before, the least-action principle is a way to formulate equations of motion in terms of a variational principle, with the Euler-Lagrange equations being these equations of motion. One of its advantages compared to the naive statement of these equations of motion is that symmetries are simpler to find and analyze for a such given equation of motion. In modern physics it's even the other way around: You try to find equations of motion obeying a given symmetry, and that's why one tries to formulate it in terms of the least-action principle with an action invariant under the corresponding symmetry transformation.

Of course, the Euler-Lagrange equations don't follow from a symmetry principle but from the stationarity of the action under arbitrary variations of the quantities in question (with fixed boundary conditions). Even an action without any symmetries defines equations of motion in terms of the Euler-Lagrange equations.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

An off topic subthread has been deleted and the thread has been reopened.