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What is common theme between U(1) SU(2) and SU(3)

  1. Jun 7, 2013 #1
    Hello! If all the elements of a Unitary group can be found using Euler's formula, does that mean that each unitary group represents some kind of cyclic transformation, since we are talking about a circle? I think I read that U(1) is a phase transformation, and SU(2) is a spin transformation? Is this correct? If so, what then would SU(3) describe?

    Thanks
     
  2. jcsd
  3. Jun 7, 2013 #2

    fzero

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    The unitary groups are an example of compact manifolds. There is a precise mathematical definition of compact that you can look up on wikipedia, for example, but roughly it means that you can use coordinates with finite ranges to parametrize the manifold. These coordinates can indeed be chosen as angles. U(1) is a circle, SU(2) can be identified with a 3-sphere, while SU(3) is closely related to the product manifold ##S^3\times S^5## (more precisely it is a fiber bundle of ##S^3## over ##S^5##).

    In physics, these groups actually have many different interpretations and applications. U(1) is indeed a phase transformation and SU(2) corresponds to spin. But there are other uses. SU(2) spin corresponds to a subgroup of the Lorentz group of symmetries of Minkowski space. But SU(2) is also the approximate symmetry identified with isospin, which explains that the strong nuclear interaction treats the proton and neutron in the same way. The differences are mainly a feature of the electromagnetic interaction (since the proton is electrically charged while the neutron is not) and the small difference in mass. SU(2) is also part of the gauge group of the electroweak interaction. These are very different applications, but the same mathematical structure of the SU(2) group lies behind them.

    SU(3) also has a large number of different applications. It is the gauge group of the strong interactions, but it also appears as an extension of the isospin group that includes the properties of the strange quark. However, it also appears as a symmetry of the harmonic oscillator in 3-dimensions.
     
  4. Jun 7, 2013 #3
    Very informative!!!! Thank you, fzero! I didn't realize SU(2) was so useful... so it then probably existed before gauge theory?? Did gauge theory tailor everything to fit into the Unitary Groups?

    Also, two more questions about Lie Groups:
    1. Are they a central part of quantum mechanics?
    2. Or, if not in QM, are they used in QFT to describe a force "locally" (????) Is that what is taking place? What exactly is the purpose of the Lie Algebra, in a nutshell?

    My impression is that a vector space is being created (by the generators) that has finite dimensions, but each axis is infinite? And the elements within the vector space are all matrices, which represent all the possible transformations? Is this vector space ℝ? Hilbert? Lots of questions, don't have to answer them all, though!! :smile:
     
  5. Jun 7, 2013 #4

    fzero

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    Yes, symmetry groups were already known to Euclid, c. 300 BC, but the modern approach to Lie groups only began in the 19th century.

    Not exactly. The formalism of gauge theory is more general and can be studied for any Lie group, compact or not. The presence of the unitary groups in the Standard Model is a feature of nature, an observational fact, not something that was tailored by the theory.

    Well they are not central in the sense that they are part of the fundamental postulates. However, they are certainly central to the practical description of most phenomena owing to the natural presence of a symmetry that is of the Lie type. This is most often a result of the Lorentz group, which has the rotation group as a subgroup.

    Again, there are many ways in which Lie groups appear. The Lorentz group is one, but this is not usually considered a "local" symmetry. When we apply a rotation to a system, we do it the same way at all points. It is usually viewed as a way to describe how the physics is the same regardless of what coordinate system we choose. This is an example of a "global" symmetry. In a gauge symmetry, we can consider transformations which depend on a particular point of spacetime, which are considered "local" transformations.

    The purpose of a global symmetry group is usually to identify simplifications in our description of a system. For instance, if we note that a central force is spherically symmetric, we can see that the main ingredient in the description of an object experiencing the force is the distance from the source, rather than the angular position in space.

    The purpose of a local, or gauge, symmetry is similar, but a bit different. We say that a gauge symmetry indicates that some of the degrees of freedom that we use to describe the system are redundant. They are there to make the equations simpler, but end up being eliminated by a constraint, which is the choice of gauge.

    I think it's difficult to really explain this without additional QFT background, but perhaps a simple example might illustrate something useful. Imagine that we have a bead sliding on a circular wire. This superficially looks like a two-dimensional problem, so in classical mechanics we would have as our degrees of freedom, the ##x## and ##y## positions of the bead, as well as the conjugate momenta ##p_x, p_y##. However, the fact that the bead is confined to the wire can be viewed as a constraint on the coordinates: ##x^2 + y^2 = R^2##. This equation tells us that one position coordinate is actually redundant, i.e. it can always be determined if we know the other one. Furthermore, there is an corresponding redundancy in the conjugate momenta, since the linear combination of ##p_x, p_y## corresponding to the radial momentum must always be zero. So the true degrees of freedom are the angular position of the bead and its angular momentum.

    In gauge theories, we have something similar. We start with the degrees of freedom corresponding to a gauge field, which can be viewed as a collection of matrices (with size corresponding to the particular group). But the gauge symmetry tells us that some of these degrees of freedom are not independent from the others. The difference with the classical mechanics example above is that the system of equations for the relevant variables is much harder to solve than the system for the redundant set of variables. So it is an easier mathematical problem to consider the extra degrees of freedom and correctly account for the constraints in the process of solution.

    If we were discussing a quantum mechanical system, then the states will live in some Hilbert space, which is a generalization of the notion of a vector space, where we allow the elements of the vectors to be certain well-defined functions. This depends on the particular system. If we were discussing a spin system, then the state space is usually a normal vector space. For example, a spin 1/2 system has a basis of spin up and down states

    $$ | + \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} , ~~~ | -\rangle = \begin{pmatrix} 0 \\ 1\end{pmatrix}.$$

    As the system becomes more complex, so does the state space.

    Now the symmetry group is something that acts on the elements of the state space. For the spin 1/2 system above, it can be represented by 2x2 unitary matrices acting on the 2d vectors that span the state space.

    Things can get more complicated very fast. For example, certain QM problems have states that should be described as spherical harmonics, which are genuine functions. The rotation group acts on these as differential operators, rather than as ordinary matrices. However, we can assemble the spherical harmonics into vectors whose components correspond to the harmonics with specific values of the quantum numbers ##\ell, m##, then the rotation generators will have a realization as matrices again. In general, you should always think of the states as "vectors," then the generators of the symmetry group are to be thought of as matrices that act on these vectors, though one might have to be a bit elaborate to organize it in this way.
     
  6. Jun 7, 2013 #5
    Wow, fzero... can I pay you to join me at school this summer??????!!! :!!)
    Thank you so much for your time!!!!!! And insight!!! I am extremely grateful!!!


    The presence of the unitary groups in the Standard Model is a feature of nature, an observational fact, not something that was tailored by the theory.... That is pretty amazing!

    Thanks especially for your explanation to 2...
     
    Last edited: Jun 7, 2013
  7. Jun 8, 2013 #6
    There is another thread somewhere around here where the discussion of GR as a theory with a local symmetry group (the lorentz group) is being discussed. It's in SR/GR somewhere... But the Lorentz group (without translations) is compact and and can certainly be a local symmetry group.
     
  8. Jun 8, 2013 #7

    fzero

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    It is true that there are various attempts to study quantum gravity by considering the Lorentz group as a local symmetry. These are all rather speculative, for various reasons, and also amount to an exceedlingly tiny percentage of the literature on the Lorentz group.

    Actually only the rotational subgroup of the Lorentz group is compact. Boosts correspond to noncompact transformations. It turns out that it isn't strictly necessary for the gauge group to be compact. However, a compact gauge group is sufficient to ensure that there are no negative norm (ghost) gauge boson states. For a given noncompact group, this is no longer guaranteed, but there are plenty of examples of unitary gauge theories with noncompact gauge group (##G=\mathbb{R}## should be fine, for example).
     
  9. Jun 8, 2013 #8
    True, but I was just talking about normal GR. GR can be cast into a gauge theory using tetrads and the spin connection, which has a local lorentz group symmetry.
     
  10. Jun 8, 2013 #9
    my bad about the non-compactness, I skipped a beat.
     
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