Understand Wigner-Eckart Theorem & Dimensionality of Vectors

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In summary, the Wigner-Eckart theorem states that, given a vector space V and a symmetry group G on it, with the representation U(G), we have irreducible tensors O_i^mu (where n_mu is the dimension of the irreducible representation of G) that transform under the direct product representation D^(mu x nu). By using Clebsch-Gordan coefficients, we can decompose the vector space into invariant subspaces and obtain the basis vectors |w^lambda_alpha_l> corresponding to the irrep D^lambda. The Wigner-Eckart theorem is then obtained by calculating the product of <e^l_lambda| and O_i^mu |e_j^nu>, which may have different dimensionalities since
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Hello! I am a bit confused about the dimensionality of the vectors in Wigner-Eckart theorem. Here it is how it gets presented in my book. Given a vector space V and a symmetry group on it G, with the representation U(G) we have the irreducible tensors $${O_i^\mu,i=1,...,n_\mu}$$ (where ##n_\mu## is the dimension of the ##D^\mu## irreducible representation (irrep) of G) having the property that $$U(g)O_i^\mu U(g)^{-1}=O_i^\mu D^\mu(g)^j_i$$ for all ##g\in G##. Now for a set of irreducible tensors ##O_i^\mu## and a set of orthonormal vectors ##e_j^\nu##, vectors which span an invariant subspace of V, we have: $$O_i^\mu |e_j^\nu>=O_k^\mu |e_l^\nu>D^\mu(g)^k_i D^\nu(g)^l_j$$ which shows that the vectors ##O_i^\mu |e_j^\nu>## transform under the direct product representation ##D^{\mu \times \nu}##. Then using Clebsch-Gordan coefficients, we can diagonalize ##D^{\mu \times \nu}## and implicitly decompose the vector space it acts on into invariant subspaces. Thus we get: $$O_i^\mu |e_j^\nu>=\sum_{\alpha,\lambda,l}|w^\lambda_{\alpha l}><\alpha,\lambda,l(\mu \nu)i,j>$$ where ##<\alpha,\lambda,l(\mu \nu)i,j>## are the CG coefficients and ##|w^\lambda_{\alpha l}>## are the basis vectors corresponding to the irrep ##D^{\lambda}## in the decomposition of ##D^{\mu \times \nu}##. Lastly, to get the Wigner-Eckart theorem they calculate this product: $$<e^l_\lambda|O_i^\mu |e_j^\nu>$$ and here it is where I get confused. The vector (actually I think it becomes a one form, but anyway) ##<e^l_\lambda|## is in the vector space V (it has lot's of zeros as it is invariant under a certain subspaces, but still in V). However ##O_i^\mu |e_j^\nu>## is in the vector space associated with ##D^{\mu \times \nu}##. Now the first one would have the dimension the same as V while the second one will have a dimension ##n_\mu \times n_\nu## which don't have to be equal. How can you take the dot product of 2 vectors with different dimensionalities? What am I missing here? For reference the book I am using is Group Theory in Physics by Wu-Ki Tung and this is presented towards the end of chapter 4. Thank you!
 
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1. What is the Wigner-Eckart theorem?

The Wigner-Eckart theorem is a mathematical theorem in quantum mechanics that allows for the simplification of matrix elements of certain operators between states of different angular momentum. It is named after physicists Eugene Wigner and Carl Eckart.

2. How does the Wigner-Eckart theorem relate to the dimensionality of vectors?

The Wigner-Eckart theorem is closely related to the dimensionality of vectors because it allows us to decompose a vector into its components in different subspaces. This means that we can represent a higher-dimensional vector in terms of lower-dimensional vectors, making complex calculations more manageable.

3. What is the significance of the Wigner-Eckart theorem in quantum mechanics?

The Wigner-Eckart theorem is of great importance in quantum mechanics as it provides a powerful tool for simplifying calculations and understanding the relationship between different states and operators. It has numerous applications in fields such as atomic and molecular physics, nuclear physics, and solid-state physics.

4. Can the Wigner-Eckart theorem be applied to all types of vectors?

No, the Wigner-Eckart theorem is specifically applicable to vectors in quantum mechanics that represent physical states with different angular momentum. It cannot be directly applied to vectors in other contexts.

5. Are there any limitations or assumptions associated with the Wigner-Eckart theorem?

Yes, the Wigner-Eckart theorem has certain limitations and assumptions. It only applies to systems with angular momentum and it assumes that the underlying symmetry is spherical. Additionally, the theorem is only valid for systems with a finite number of degrees of freedom and does not apply to continuous systems.

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