Tagged Questions

Use as a synonym to the representation-theory tag

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Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
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How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
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Unitarily Inequivalent Representations

The definition of unitarily equivalent representations I am using is the one given here: https://en.wikipedia.org/wiki/Haag%27s_theorem. Now in this text http://www.sa.infn.it/Massimo.Blasone/...
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Double groups in Crystallography

I'm currently studying double point groups and their applications in condensed matter physics. Let me start by giving you the definition of the double group that is used in my textbook: Let $G$ be a ...
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How unique are the quantum numbers we commonly use?

We use the eigenvalues of the Cartan generators (=diagonal generators) of a given gauge group as quantum numbers in physics. Are these numbers somehow fixed and if not, what transformations are ...
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What does it mean by saying the generators of translations transform as vectors under the Lorentz Group?

The commutator of generators of Lorentz transformation and translation is as follow: $$[M^{\mu\nu},P^\sigma]=i(P^\mu\eta^{\nu\sigma}-P^\nu\eta^{\mu\sigma} ).$$ Then from this we usually say that the ...
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On the Lorentz Group representation [closed]

I am going through the notes on QFT by Srednicki (which is certainly a worth reading on the subject, and can be found online, see http://web.physics.ucsb.edu/~mark/qft.html). When describing fermions,...
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Degenaracy in mass of $8$ and $27$ reps of $SU(3)$ in Coleman's Aspects of Symmetry [closed]

In Coleman's Aspect of symmetry he proposes an amusing problem in the first chapter. It asks us to consider a set of eight pseudo-scalar fields transforming in the adjoint representation of $SU(3)$. ...
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How to check if some term in the Lagrangian involving gauge bosons is gauge invariant without explicit computations?

Normally (for fermions and scalars) we can simply use the decomposition of tensor products of gauge group representations to find invariant terms that we can write into the Lagrangian. For example ...
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Do gauge bosons really transform according to the adjoint representation of the gauge group?

Its commonly said that gauge bosons transform according to the adjoint representation of the corresponding gauge group. For example, for $SU(2)$ the gauge bosons live in the adjoint $3$ dimensional ...
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Rest Mass and Wigner's Classification

I believe (but please correct me if I'm wrong) that I understand the basic philosophy and most of the mathematics involved in Wigner's classification of particles via group representations. But I'm ...
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Representation of the Standard Model group $SU(3) \times SU(2) \times U(1)$

As the gauge group of the Standard Model is $SU(3) \times SU(2) \times U(1)$, would the associated fermions fields be the product of a triplet, a doublet and a singlet, for all particles, or is that ...
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How is the Full Standard Model group representation displayed?

I have often seen, on YouTube lectures and textbooks, the direct product gauge group representation listed below and it is often accompanied with a statement to the effect that "this is how we sum ...
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Invariant linearly independent scalar potential construction for product groups

Lets say one has a gauge group for example SU(n) or SO(n) and has a scalar field which belongs to a certain representation (m-ranked tensor). If one wants to write down the invariant potential for the ...
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Do particles have spin because there exist spinor representations for the Lorentz group?

I am reading Peskin and Schroeder's An introduction to field theory. They first describe the spinor representation of the Lorentz group, and then they mention the fact that different particles have ...
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Representation of U(1) on fock space

I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ...
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Traceless Tensors in $SU(3)$, Georgi's Lie Algebras

I'm doing a self-study through Georgi's Lie Algebra's in Particle Physics and there is a ''note without proof'' in the book that I have not managed to see through myself. In Section 10.3, Georgi ...
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Let $$u=\left( \begin{array}{cccc} c_1&c_2&c_3&c_4 \end{array} \right)^T$$ for $$\psi = c_1\psi_1 + c_2\psi_2 + c_3\psi_3+ c_4\psi_4$$ We assume that $\left<\psi_i|\psi_j\right> = \... 3answers 528 views Explaining a quote by Weinberg about the signifcance of symmetry groups in physics When skimming through a book, I found this quote: The universe is an enormous direct product of representations of symmetry groups. —Steven Weinberg I am a mathematician (so I know only basic ... 1answer 76 views Getting to spins of arbitrary direction Let me rephrase this question: Let us assume we know that symmetry transformations always look like this: $$U(s)=e^{iKs}$$ with a hermitian Operator K. This tells us that for very small$s$: $$U(... 1answer 81 views Representations of Lorentz algebra It is well known that the Lorentz algebra can be written as two SU(2) algebras. By defining$$N_i=\frac{1}{2}(J_i+iK_i), \qquad N^{\dagger}_i=\frac{1}{2}(J_i-iK_i)$$we have [N_i,N_j]=i\... 0answers 37 views symmetry group of multi-electron atom Neglecting spin effects, the energy levels of multi-electron atoms are characterized by states of definite total orbital (L^2) and spin angular momentum (S^2). From this it seems that the ... 2answers 238 views Where does the Lorentz boost for a Dirac spinor come from? I have read, that if you have a Dirac spinor $$\psi = \begin{pmatrix} \phi_R\\ \phi_L \end{pmatrix}$$ that you can apply a Lorentz boost along the z-direction with ... 0answers 88 views Decomposition of a tensor under transformations To illustrate my question I'll take an example from theory of relativity: An arbitrary 4-tensor A^{ik} changes under a general coordinate transformation:$$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$(... 2answers 183 views Why do decompositons like 16 \otimes 16 = 10 \oplus 120 \oplus 126 tell us which Higgs representations we can use? EDIT: I found an answer, which I do not understand: In Gürsey - Symmetry breaking patterns in E6 he writes: " Because of Fermi-Dirac statistics of fermions they must occur in the symmetric part of ... 1answer 815 views Spin commutation relations For orbital angular momentum defined as L= r \times p we can prove, in quantum mechanics, the commutation relations. Also, we could prove these relationships through the study of rotations (... 3answers 73 views Uniqueness of expression of a Lie group element Just take the SU(2) group as an example. The three generators are J_z, J_+, and J_-. For an element g , sometimes we want to express it as$$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} .$$... 1answer 264 views Why$SU(3)$has eight generators? The generators of$SU(3)$group are Gell-Mann matrices and one can construct these generators from Pauli spin matrices, basically expanding in 3d and rotating about each axis. Take$\sigma_3\$, assume ...
I have a Hamiltonian for 3 particles of spin 1 that I boiled down to: $$k(\textbf{S}^2+\cdots),$$ where: \textbf{S}=\textbf{S}_1+\textbf{S}_2+\textbf{S}_3. ...