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Lorentz group, Poincaré group and conformal group

  1. Feb 12, 2007 #1
    Dear all,

    I just received by mail the book by Weinberg.
    I am very very happy. At each page I can see something new to learn.

    But I would like to learn a bit more about his remark on page 28.
    (you can read it with the amazon reader)
    He talks about the 10 parameter Poincaré group, that I knew already.
    But he also mentions a 15-parameter group called "conformal group".
    He explains this is the group of transformations that leave invariant the ds²=0 condition (the ds² is not necessarily invariant, only the condition ds² is).

    I would learn to much more about that.
    For example, Weinberg excludes this group as a valid symmetry in physics, but the explanation is not totally clear for me.
    Also, why are there 15 parameters in this group?
    Also, what do these transformation look like?
    Also, when I read Landau-Lifchitz, I read arguments explaining that in any symmetry tranformation we should have ds²=A(V)ds'² and that A(V) should be a constant to satisfy the hypothesis of space homogeneity. I did not care too much at that time. But now, with this new reading in Weinberg, I need to understand more and see the missing link.

    Thanks for your impetus ...

    Last edited: Feb 12, 2007
  2. jcsd
  3. Feb 12, 2007 #2
    In the BUSSTEPP lecture notes on SUSY here the author mentions an extension to the Poincare algebra into the Conformal algebra on Page 13. The new generators are [tex]K_{\mu}[/tex] and [tex]D[/tex] which represent the new tranformations. Coupled with the 10 original Poincare generators, that makes 15.

    Reading around those pages in the notes should help explain what they do.
  4. Feb 12, 2007 #3


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    Yea basically the conformal group is the largest possible group that leaves the mass condition m^2-p^2=0 invariant (likewise with SUSY, the superconformal group does the same) and that isn't trivially modable out by gauge redundancies. Since its bigger than the poincare group, it is of course of interest. As a general rule, more symmetry = easier to solve problems in a tractable manner.

    Typically what people do is study the group than look at ways to spontaneously break it down to the poincare group, and then analyze how and where thats possible and what goes wrong etc. A tremendous amount of insight in modern field theory has come from studying these sorts of objects.

    Anyway, its useful to remember that it doesnt really act on minkowski spacetime, but rather a sort of abstract completion thereof (callled the conformally complete compactification). But im rambling...

    When Weinberg says it cannot describe physics, what he means is that the *full* theory of course cannot have pure scale invariance (eg we have length scales, usually set by some mass) ergo the conformal group has to be badly broken in some way. However what he doesnt say is that it turns out that the YangMills equation typically has (under some conditions) an invariance under this symmetry group. Which is beautiful. So in some sense a certain sets of physical entities (fields), when we can treat them in isolation (usually in thought experiments), then this is the right group to be using.
  5. Feb 17, 2007 #4
    I would like to "see" one member of the conformal group which is not part of the Poincaré group.
    What would the coordinate transformation look like?
    Weinberg says that these members should be nonlinear transformations.
    I can't see an example.
    Once I know an example, I could maybe understand easier why these transformations are described by (only) 15 parameter, 10 for the poincaré degrees of liberty and 5 more for these other members.

    Thanks to show me an example,

    Last edited: Feb 17, 2007
  6. Feb 17, 2007 #5


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    One example is a scaling transformation [tex] \vec r' = k \vec r [/tex], where [tex]k[/tex] is a constant. This transformation doesn't have determinant 1.
  7. Feb 17, 2007 #6
    Thanks robphy!
    I am not sure if members of the conformal group should have det(M)=1.
    But I would guess yes because rescaling is not incompatible with the laws of physics.
    Weinberg says that physics excludes the proper members of the conformal group as a symmetry of physics.
    Weinberg also says that these tranformations are nonlinear.
    One example of these would make my day.


  8. Apr 19, 2007 #7


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    Conformal Group Representation

    I noticed the old thread and though I'd post a follow-up.

    Here is how I "play" with the conformal group. First imagine the action of SO(2,4) on a 2+4 dimensional space. Label the six ortho-normal coordinates: [tex](t,u;w,x,y,z)[/tex].
    With t,u "time-like" and the remainders space-like. Now since the conformal group only generates homogenous transformations we must view its point action in terms of translations (Lie vector fields) on one of the pseudo-spheres:

    [tex]t^2+u^2 - (w^2+x^2+y^2+z^2) =[/tex] one of {0,1,-1}

    Anyway, I prefer just the think in terms of rays in this 6-space as the "points".

    Here is the neat thing. Imagine the subset of these pseudo-rotations which leaves a given vector, S invariant. If S is a time-like vector then you get the sub-group SO(1,4) (deSitter) and if S is space-like you get the sub-group SO(2,3) (anti-deSitter), but if S is null then you get the subgroup ISO(1,3) (Poincare!).

    So start with the null cone:

    [tex]t^2+u^2 - (w^2+x^2+y^2+z^2) = 0[/tex]

    Rewrite this as:

    [tex]t^2 - (x^2+y^2+z^2) = (w-u)(w+u).[/tex]

    Letting [tex]s=w-u[/tex], and [tex]s'=w+u[/tex], note that these are null coordinates.
    We define the Poincare group to be the sub-group leaving [tex]s = w-u[/tex] invariant.
    Its generators are:

    Boosts: [tex]B_x = x \partial_t + t \partial_x[/tex], [tex]B_y= y\partial_t + t\partial_y[/tex], etc.
    Rotations: [tex]R_z= y\partial_x - x\partial_y[/tex] etc.
    Translations: [tex]P_x= s\partial_x - x\partial_{s'}[/tex], ...[tex]P_t=s\partial_t - t\partial_{s'}[/tex].

    Note that: [tex][P_x,x/s] = [s\partial_x-x\partial_{s'}, x/s] = 1 [/tex]
    so we choose as space-time coordinates:
    [tex](T,X,Y,Z) = (t/s, x/s, y/s, z/s)[/tex]
    Note then that the proper time/(neg. proper distance) for the coordinate point from the origin is:
    [tex]T^2 - X^2 - Y^2 - Z^2 = s'/s = \tau^2[/tex].

    Now the remaining 5 generators are:
    [tex]C_x = s'\partial_x - x\partial_s [/tex]
    and similarly with[tex]C_y, C_z, C_t[/tex]
    [tex]C_s = s'\partial_{s'} - s\partial_s[/tex].

    Cs is the global scaling generator:
    [tex] [C_s,X]= [-s\partial_s,x/s] = x/s = X[/tex] and likewise with Y,Z,T.

    The others yield non-linear transformations for example:
    [tex][C_x,X] = s'/s(1+X)[/tex] which is a mixture of scaling and translation in the X direction but depending on the value of [tex]s'/s=\tau^2[/tex].
    [tex][C_x,Y] = [-x\partial_s,y/s] = xy/s^2 = XY [/tex] etc. {error (partially) corrected!}

    [TODO: Rework in terms of standard presentation of conformal transformations, and check errors.]
    Last edited: Apr 19, 2007
  9. Apr 19, 2007 #8

    Chris Hillman

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    Explicit generators for so(2,4) = conf(1,3)

    Michel, actually I cannot use Amazon reader and don't have access to my p.c. right now, but I assume Weinberg is discussing a certain finite dimensional subgroup of the point symmetry group of the three-dimensional wave equation, namely the conformal group on Minkowski spacetime Conf(1,3).

    Michel, the Lie algebra of this Lie group can be thought of as vector fields on Minkowski spacetime and they can be determined by an algorithm given by Lie (see the book by Olver on symmetries of PDEs). The result is a generating set of 15 vector fields:
    1. Translations:
    X_1 = \partial_x, \; X_2 = \partial_y, \; X_3 = \partial_z, \; X_4 = \partial_t
    2a. Boosts:
    X_5 = t \, \partial_x + x \, \partial_t, \;
    X_6 = t \, \partial_y + y \, \partial_t,
    X_7 = t \, \partial_z + z \, \partial_t,
    2b. Rotations:
    X_8 = y \, \partial_x - x \, \partial_y, \;
    X_9 = z \, \partial_y - y \, \partial_z, \;
    X_{10} = x \, \partial_z - z \, \partial_x
    3. Dilation:
    X_{11} = x \, \partial_x + y \, \partial_y + z \, \partial_z + t \, \partial_t
    4. Inversions (Moebius transformations; ignore the [itex]\partial_u[/itex] terms for the moment):
    X_{12} = \left( x^2-y^2-z^2+t^2 \right) \, \partial_x
    + 2 \, x \, y \, \partial_y + 2 \, x \, z \, \partial_z + 2 \, x \, t \, \partial_t
    - 2 \, x \, u \partial_u
    X_{13} = \left( -x^2+y^2-z^2+t^2 \right) \, \partial_y
    + 2 \, x \, y \, \partial_y + 2 \, x \, z \, \partial_z + 2 \, x \, t \, \partial_t
    - 2 \, x \, u \partial_u
    X_{14} = \left( -x^2-y^2+z^2+t^2 \right) \, \partial_z
    + 2 \, x \, y \, \partial_x + 2 \, y \, z \, \partial_y + 2 \, z \, t \, \partial_t
    - 2 \, z \, u \partial_u
    X_{15} = \left( x^2+y^2+z^2+t^2 \right) \, \partial_t
    + 2 \, x \, t \, \partial_y + 2 \, y \, t \, \partial_y + 2 \, z \, t \, \partial_z
    - 2 \, t \, u \partial_u
    The grouping is meant to suggest various interesting subalgebras, such as the group of "Minkowski similitudes" (11 dimensional), the Poincare group (10 dimensional), the Lorentz group (6 dimensional), rotation group (3 dimensional), etc.

    To find the corresponding one-parameter or unidimensional subgroups of the Lie group itself, solve the appropriate system of ODEs to find the flow generated by each of these vector fields (or even, any linear combination of them--- in principle one can write down an expression for an arbitrary element of the group this way although I wouldn't recommend trying to do that unless you have a good reason!).

    Example: the vector field [itex]-y \, \partial_x + x \, \partial_y[/itex] gives the system
    [tex] \dot{x} = -y, \; \dot{y} = x, \; \dot{z} = 0, \; \dot{t} = 0[/tex]
    which you can solve to find
    [tex] x(s) = x_0 \, \cos(s) - y_0 \sin(s), \;
    y(s) = x_0 \, \sin(s) + y_0 \, \cos(s)
    Here, s is the parameter of the unidimensional subgroup, whose orbits are simply circles which keep z,t constant.

    The extra terms in the inversions come from the fact that when we consider this group to act on the independent variables [itex]x,y,z,t[/itex] and dependent variable [itex]u[/itex] in the three-dimensional wave equation [itex]u_{tt} = u_{xx}+u_{yy}+u_{zz}[/itex], the inversions act on both independent and dependent variables. The Lie algebra of the point symmetry group itself contains some more generators due to the linearity of this equation. In fact, infinitely many, but the extra ones aren't very interesting.

    It's a good exercise to follow the procedure sketched above for the inversions (with the [itex]\partial_u[/itex] terms deleted) to see what they look like. Likewise, to determine the orbits.

    One fun thing you can do with this kind of analysis is determine the subgroup of variational symmetries. This is useful because by Noether's principle each of these are associated with a locally conserved quantity. For a solution which decays sufficiently rapidly, via Stoke's theorem one may obtain invariants of the solution, such as energy (from time translation), momentum (from the spatial translations), angular momentum (from the rotations), plus somewhat more exotic things (from the boosts).

    Another fun thing you can do is find solutions having specified symmetries. For example, you might seek a solution of the wave equation which is invariant under a certain group of parabolic Lorentz transformations like [itex]x \, \partial_t + (t-y) \, \partial_x + x\, \partial_y[/itex]. Cranking through the method, one finds in this case
    [tex] u(t,x,y) = G(y-t) + \frac{F \left( \frac{x^2+y^2-t^2}{y-t} \right)}{\sqrt{y-t}} [/tex]
    (in the two-dimensional case), where G is an arbitrary smooth function.

    (Arghghgh, I said a lot more but the system lost my work...)
    Last edited: Apr 19, 2007
  10. Apr 19, 2007 #9


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    Yea, I realized this formum allows latex and have been revising my post.

    W.r.t. your (standard) exposition, Let me fiddle a bit and see if I can map between them. (and maybe catch an error).

    The main connection should manifest as a convention for projecting (deformation contracting) the rays in the 2+4 dimensional
    space (call it conformal space?) onto the points of minkowski space-time (and/or deSitter anti-deSitter space-time).

    Last edited: Apr 19, 2007
  11. Apr 19, 2007 #10

    Chris Hillman

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    Reconciling standard vs. jambaugh's approach

    Your u is not the same as my u, as you no doubt realized!
  12. Apr 20, 2007 #11
    Page 28 and 29 of the above mentioned book.
    Last edited: May 20, 2007
  13. Apr 26, 2007 #12


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    Right Chris,
    I think
    [tex]u = exp(s')[/tex]
    for your u and my s'.
    Also I lost one of my corrections:
    [tex] [C_x,X]= \tau^2(1+X^2)[/tex]
    and so
    [tex]C_x[/tex] corresponds (I think) to a linear combination of [tex]X_{12}[/tex] and [tex] X_{11}[/tex].
    Similarly for the other four inversions.

    Finally I think I was not careful enough with signs in some of the generators.
    This doesn't affect the too much in the given exposition. But my "Poincare generators" may not be quite right.

    I'll fiddle with this some more after the semester here is over.

  14. Jun 9, 2007 #13


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