# Tag Info

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You should also specify the Representation. The Representation requires SU(N) Lie group with N×N matrix is called Fundamental Representation. Which is used in Standard model U(1) x SU(2) x SU(3). You can surely have SU(N) Lie groups with other Representation, such as Adjoint Representation, then in this case SU(N) are represented by a matrix with a rank of ...

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The way I understand it, there is actually no shortcut method to write the explicit tensor components from the Young tableaux. We just need to symmetrize and antisymmetrize in all upper and lower indices, and whenever we come across mixed indices, remove traces as well. Traceless symmetric tensors, antisymmetric tensors and traces transform irreducibly. To ...

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In general, the tensor product of tensors with both being in the same representation (e.g. fundamental) results in a decomposition to symmetrical and antisymmetrical parts. This can be seen that for example from the fact that the expression you wrote down is equal to (after rewriting the epsilon-tensors in terms of Kronecker-deltas) ...

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Not all irreducible representations (irrep's for short) of the Poincaré group lead to a Lagrangian. One example (see my comment to Julio Parra's answer) are the zero-mass, "continuous-helicity" (sometimes called "infinite-helicity") representations. There is, however, a way to begin from a positive energy irrep of the Poincaré group (i.e. a 1-particle ...

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Alex Nelson's answer is much better that mine, but it doesn't address your question at all. (a) Does it form a group? No, it doesn't. See bellow, to find out what does 'look like' (but isn't) a group. (b) What are the elements of the group? The group-like structure is the following. Being sloppy, effective action satisfies the folowing semigroup ...

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I'm not sure such a thing exists. Usually reps only helps you classify the kind of particles you have (i.e the quantum numbers that identify them) and how they transform under the corresponding group. I believe how to represent this particles mathematically and what is their dynamics is a different matter. The only thing similar I know about is that some ...

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There are really several questions here: (a) What is the renormalization group? Specifically the law of composition, etc. (b) How does the equation the OP gave relate to this? Short Answer It's a semigroup (see references below). The equation you wrote, $$\tag{1} \left[\mu \frac{\partial}{\partial \mu} + \beta \frac{\partial}{\partial g} + m \gamma_{m^2} ... 1 There is a recent textbook which gives a fairly complete and concise presentation of group theory, covering both structure and representations of both finite and continuous (Lie) groups, with a brief discussion on applications to music (finite groups) and elementary particles (Lie groups). The target level is advanced undergraduate and beginning graduate. It ... 4 An angular momentum eigenstate can be rotated using, $$\left| J , m \right\rangle \rightarrow e ^{ i {\vec S} \cdot {\vec \theta} } \left| J , m \right\rangle$$ where  {\vec S}  is the 2J+1 dimensional Pauli matrices. For spin  1/2  for example,  {\vec S}  are just the ordinary Pauli matrices,  \frac{1}{2} ... 0 Electric charge conservation is a "discrete" symmetry. Quarks and anti-quarks have discrete fractional electric charges (±1/3, ±2/3) electrons, positrons and protons have integer charges. 1 Below follows the proof which Howard Georgi seems to have in mind. Let us call the root vector(s) in the Dynkin diagram (a) corresponding to the single 3-vertex for \vec{\gamma}, the three 2-vertices for \vec{\beta}_1, \vec{\beta}_2, \vec{\beta}_3, and the three 1-vertices for \vec{\alpha}_1, \vec{\alpha}_2, \vec{\alpha}_3. Since ... 4 Essentially by definition (due to Wigner), one-particle Hilbert spaces of elementary particles support unitary strongly continuous irreducible representations of Poincaré group. Conversely, any multi-particle Hilbert space, with either fixed or undefined number of particles either identical or distinguishable, cannot be irreducible under the action of the ... 2 Good Lord! Is Schwinger still worth reading? A top physicist of course, but unfortunately right over my head! You're talking about the big A Adjoint representation as in Hunter's Answer, and there are a great many more modern, rigorous and way clearer texts on this topic. The Wikipedia page is a great start. Also see Rossmann, "Lie Groups, An Introduction ... 2 As I mentioned in my comment, I believe you are talking about the adjoint representation of a Lie Group G with a Lie algebra \mathfrak{g}: $$\forall x \in \mathfrak{g}, \;\; \mathrm{Ad} \: D(g) : x \mapsto D(g) x D^{-1}(g) \in \mathfrak{g}$$ where D(g) denotes a represention of g \in G. One method you can see why D(g) x ... 1 You may know already that "symmetry" is not always important when it comes to non-Euclidean spaces. For instance, in quantum mechanics, a symmetric operator is seldom important, but one that is equal to its Hermitian conjugate--one that is Hermitian or "self-adjoint"--is incredibly important, for those operators have real eigenvalues and thus correspond to ... 2 Indeed NowIGetToLearnWhatAHeadIs's comment answers your question: "Simply because I hadn't encountered one that was not." A rotation is a lorentz transformation which is not symmetric. Indeed the transpose of a rotation matrix is its inverse, and only trivial rotations or rotations through half a turn are involutary (self inverse). To see this in ... 2 You've got two very good answers from Hunter and NowIGetToLearnWhatAHeadIs. However, it's probably useful to know that this beast O(1,3) is isomorphic or locally isomorphic (i.e. has the same Lie algebra) to a surprising number of other interesting groups, which each give you a slightly different way to think about it. First note that its identity ... 3 It's the same way you know there are three parameters in SO(3). The equation \Lambda^T \eta \, \Lambda = \eta has (n^2+n)/2 independent scalar equations. To see this, write the equation in component form: \Lambda^{\mu\nu} \Lambda_\mu{}^\rho = \eta^{\nu\rho}. Now we see there are n^2 scalar equations equations, but because \eta is symmetric and ... 7 From special relativity we know that a Lorentz transformation: $$x'^\mu = \Lambda^\mu {}_\nu x^\nu$$ preserves the distance: $$g^{\mu \nu} \Delta x_\mu \Delta x_\nu = g^{\mu \nu} \Delta x_\mu' \Delta x_\nu'$$ The above two equations imply: g^{\mu \nu} = g^{\rho \sigma}\Lambda_\rho ... 1 The topological manifold of the Lorentz group can be continuously embedded in the metric space \mathbb{R}^{16} together with (metric) topology inherited from \mathbb{R} (direct product topology). The subset of Lorentzian boosts in 1 spatial direction can be parametrized by \beta =v/c and is hence homeomorphic as a topological space with the open unit ... 8 I) Firstly, we are talking about the direct or Cartesian product SU(2)\times SU(2) of groups, not the tensor product^1 SU(2)\otimes SU(2) of groups. II) Secondly, SU(2)\times SU(2) is not isomorphic to the Lorentz group SO(3,1) but rather to a compact cousin$$[SU(2)\times SU(2)]/\mathbb{Z}_2~\cong~ SO(4).$$In particular, a ... 4 The problem here is with the identification of the (A,B) values of a representation with spin. A and B do not correspond to spin (they are not even Hermitian!), they just happen to obey SU(2) Lie algebras, and as such they add up in the same way that spins do. When we say that A_\mu,J_\mu,p_\mu,... are all in the (\frac{1}{2},\frac{1}{2})  ... 5 General Remarks. In general, you cannot "derive" a representation of a given group G on the objects you're considering, but there are some really standard definitions of certain group representations which are given special names like "scalar," "vector," and so on. However, given the representation of a Lie group G, this induces a representation of its ... 10 On the actual Hilbert space of a consistent relativistic quantum mechanical system, the Lorentz transformations including boosts actually are unitary – which also means that the generators J_{0i} are as Hermitian as the generators of rotations J_{ij}. We say that the Hilbert space forms a unitary representation of the Lorentz group. What the OP must be ... 4 Every Lie group has an adjoint representation. I'm not sure what definition you come at the adjoint representation from, but here's the fundamental one which I'm sure you'll see is always meaningful. Think of a C^1 path \sigma:[-1,1]\to\mathfrak{G} through the identity in a Lie group \mathfrak{G} with \sigma(0) =\mathrm{id} and with tangent X ... 5 What is a physical theory/model? A given physical theory is typically mathematically modeled by some set \mathscr O of mathematical objects, and some rules that tell us how these objects correspond to a physical system and allow us to predict what will happen to that system. For example, many systems in classical mechanics can be described by a pair ... 2 Symmetries indeed have a broad and powerful impact in physics, and I will only be able to scratch the surface of the subject in this answer, but I will try to give you a glimpse of the subject. In the most simple framework, you mention an electrostatic problem. In such a problem, the key factor is the geometric symmetries which apply to the charged ... 6 Symmetry is present when something x doesn't change under some transformation T:$$T(x)=x In an infinite cylinder, there is radial symmetry because if you move at constant height and radius, you see the same figure. In the Lagrangian case, if you change coordinates, the Lagrangian doesn't change. $L(x') =L(x)$ In group theory, group elements will ...

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