An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee

By Nadir Jeevanjee

An creation to Tensors and staff thought for Physicists offers either an intuitive and rigorous method of tensors and teams and their position in theoretical physics and utilized arithmetic. a selected goal is to demystify tensors and supply a unified framework for realizing them within the context of classical and quantum physics. Connecting the part formalism wide-spread in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings could be a great addition to the literature on tensors and staff theory. Advanced undergraduate and graduate scholars in physics and utilized arithmetic will locate readability and perception into the topic during this textbook.

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20 The Hermitian scalar product on L2 ([−a, a]) For f, g ∈ L2 ([−a, a]), define (f |g) ≡ 1 2a a −a f¯g dx. 27) Younπxcan easily check that this defines an inner product on L2 ([−a, a]), and that {ei a }n∈Z is an orthonormal set. What is more, this inner product turns L2 ([−a, a]) into a Hilbert Space, which is an inner product space that is complete. The notion of completeness is a technical one, so we will not give its precise definition, but in the case of L2 ([−a, a]) one can think of it as meaning roughly that a limit of a sequence of square-integrable functions is again square-integrable.

Vr , f1 , . . , fs ) + cT (w, v2 , . . , f1 , . . 2) and similarly for all the other arguments. This property is called multilinearity. Note that dual vectors are (1, 0) tensors, and that vectors can be viewed as (0, 1) tensors: v(f ) ≡ f (v) where v ∈ V , f ∈ V ∗ . 3) Similarly, linear operators can be viewed as (1, 1) tensors as A(v, f ) ≡ f (Av). N. 4) 39 40 3 Tensors We take (0, 0) tensors to be scalars, as a matter of convention. 1 below that the set of all tensors of type (r, s) on a vector space V , denoted Tsr (V ) or just Tsr , form a vector space.

Ei |ej ) = ±δij ) is just an orthogonal set in which the vectors have unit length. 17 The dot product (or Euclidean metric) on Rn Let v = (v 1 , . . , v n ), w = (w 1 , . . , w n ) ∈ Rn . Define (· | ·) on Rn by n (v|w) ≡ v i wi . i=1 This is sometimes written as v · w. 7 is an orthonormal basis. 18 The Hermitian scalar product on Cn Let v = (v 1 , . . , v n ), w = (w 1 , . . , w n ) ∈ Cn . Define (· | ·) on Cn by n (v|w) ≡ v¯ i w i . 7 is an orthonormal basis. Such inner products on complex vector spaces are sometimes referred to as Hermitian scalar products and are present on every quantum-mechanical vector space.

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