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## Hot answers tagged group-theory

10

You may indeed identify the generators in the way you did. However, the Lie algebras and Lie groups are different because – as quickly said by Qmechanic – you must use different reality conditions for the coefficients. A general matrix in the $SU(2)$ group is written as $$M = \exp[ i( \alpha J_+ + \bar\alpha J_- + \gamma J_0 )]$$ where $\alpha\in ... 7 The ladder operators do belong to the real Lie algebra $$\quad su(1,1)~\cong~ so(2,1)~\cong~sl(2,\mathbb{R}),$$ but they do not belong to the real Lie algebra $$su(2)~\cong~ so(3).$$ All the above real Lie algebras have complexifications isomorphic to$sl(2,\mathbb{C})$. 4 This can be explained by thinking about the coupling of fermions to the$SU(2)$weak gauge field. Let's recap what we know Weyl fermions necessarily appear in two complex representations of the Lorentz group$L$and$R$. Only fermions in the$L$representation of the Lorentz group couple to the$SU(2)$gauge field. CPT is a symmetry of the theory. Now ... 3 I'm not altogether sure what you are asking, but I suspect the following may help. To represent rotations, spins and vectors in$SU(2)$we work as follows. Rotations live in$SU(2)$. Vectors (in the physicist's sense) live in the algebra$\mathfrak{su}(2)$. The position vector$(x,\,y,z)is: $$X =x\,\hat{s}_x+y\,\hat{s}_y+z\,\hat{s}_z = ... 3 Every Hermitian traceless matrix H is in \mathfrak{su}(N) since \mathrm{Tr}(H) = 0 and so$$ \exp(\mathrm{tr}(\mathrm{i}H)) = \det(\exp(\mathrm{i}H)) = 1so \exp(\mathrm{i}H) is unitary with determinant 1, hence in \mathrm{SU}(N). The gauge field is always in the Lie algebra of the gauge group since it is introduced to cancel terms that are ... 2 Comment to the question (v1): No, such decomposition is in general not unique. E.g. the unit element {\bf 1}_{2\times 2}\in SU(2) can be written with parameters b\in 4\pi\mathbb{Z} and a=0=c. 2 I recently read a paper on the possible kinematics: http://scitation.aip.org/content/aip/journal/jmp/9/10/10.1063/1.1664490 It states that under the 3 assumptions they made, there were more then 10 possible Lie-Algebras (while they discarded one by heuristic arguments) 2 The Lie algebra \mathfrak{su}(N), viewed as a vector space of matrices, can be equipped with the following standard inner product: \begin{align} \langle X,Y\rangle = \mathrm{tr}(X^\dagger Y), \end{align} where X^\dagger Y is the matrix product of X^\dagger and Y, and \mathrm{tr} is the trace. Since X^\dagger = X for all X\in\mathfrak{su}(N), ... 2 Just a guess... The purpose is to reproduce the nice features of SU(2). With that convention, the generators of SU(2) are, in terms of Pauli matricesT^i = \frac{1}{2}\sigma^i$$So a transformation with parameters \theta_i is given by$$U=\exp\left(-i\frac{1}{2}\theta_i\sigma^i\right)$$Things get interesant when you realize that the elements of ... 2 The gauge potential is an object that, when introduced in the covariant derivative, is intended to cancel the terms that spoil the linear transformation of the field under the gauge group. Every gauge transformation g:\Sigma\to G (on a spacetime \Sigma) connected to the identity may be written as \mathrm{e}^{\mathrm{i}\chi(x)} for some Lie algebra ... 2 For orbital angular momentum, indeed, L = x\times p even as a quantum operator, see this question. When writing a ket \lvert l,m \rangle, this is meant to live in the 2l+1-dimensional space \mathcal{H}_l = \mathbb{C}^{2l+1} on which the representation of the angular momentum algebra labelled by l exists (m is the eigenvalue of the ket for ... 1 I) Perhaps it is helpful to point out that even if the physical system S has no rotational symmetry (e.g. if the system S is a 3D an-isotropic harmonic oscillator), then the Lie group G=SO(3) of rotations still has a group action G \times S \to S on the system. See also e.g. this Phys.SE post. In particular the Hilbert space {\cal H} of the system ... 1 Can these two pictures be connected in some way? Yes, that's why the Wikipedia spinor article features a picture of a Möbius strip: GNUFDL image by Slawekb, see Wikipedia The Mobius strip also features in the Mathspages Dirac's belt article where you can read that it's "reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be ... 1 There is, in a sense, a way to 'guide' oneself to the equations of motion based on the symmetries. The form of mechanics most suitable for this purpose is Hamilton's principle - the system takes a path for which the action has a stationary value for variations with fixed endpoints:$$\delta S=0S$is generally expressed as (under some parametrization of ... 1 I think the strong CP problem is still a pretty important problem. Although some people don't worry too much about these fine-tuning problems. Just like some people didn't consider the flatness problem a big deal before the inflation theory explained it. So like every problem, it's as big as you make it. It depends on how much you value naturalness... ... 1 Any linear transformation wrought on the Lie algebra of a Lie group yields a valid Lie algebra as I think you understand (the Gell Mann matrices are actually$i$times the skew-symmetric Lie algebra members), and your proposed$\lambda_3$is a linear combination of the Gell Mann matrices. The basis comprising$i$times the Gell Mann matrices does indeed span ... 1 You should have two boost generators. You have constructed one for boost in the$x$direction, but there is also one for boost in$y\$.

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