By Nadir Jeevanjee

An creation to Tensors and team thought for Physicists presents either an intuitive and rigorous method of tensors and teams and their function in theoretical physics and utilized arithmetic. a selected target is to demystify tensors and supply a unified framework for realizing them within the context of classical and quantum physics. Connecting the part formalism everyday in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings can be a welcome boost to the literature on tensors and team 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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22) Check that this is indeed an inner product. Also check that the basis {I, σx , σy , σz } for H2 (C) is orthonormal with respect to this inner product. 19 The Minkowski metric on 4-D spacetime Consider two vectors16 vi = (xi , yi , zi , ti ) ∈ R4 , i = 1, 2. The Minkowski metric, denoted η, is defined to be17 η(v1 , v2 ) ≡ x1 x2 + y1 y2 + z1 z2 − t1 t2 . 23) are often called ‘events’ in the physics literature. are, of course arbitrarily choosing the + + +− signature; we could equally well choose − − −+.

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. 21); if that conjugation was not there, a vector like v = (i, 0, . . , 0) would have (v|v) = −1 and (· | ·) would not be an inner product. 12 Let A, B ∈ Mn (C). Define (· | ·) on Mn (C) by (A|B) = 1 Tr A† B . 22) Check that this is indeed an inner product.

Note that since we raise and lower indices with an inner product and usually use orthonormal bases, the components of I when viewed as a (2, 0) tensor and when viewed as a (1, 1) tensor are the same, cf. 18. We’ll discuss I further in Sect. 6. 4 Multipole moments It is a standard result from electrostatics that the scalar potential (r) of a charge distribution ρ(r ) localized around the origin in R3 can be expanded in a Taylor series as1 (r) = 1 Q0 Q1 (r) 1 Q2 (r, r) 1 Q3 (r, r, r) + + + ··· + 4π r 2!