New PDF release: An Introduction to Tensors and Group Theory for Physicists

By Nadir Jeevanjee

ISBN-10: 0817647147

ISBN-13: 9780817647148

An creation to Tensors and crew conception for Physicists offers either an intuitive and rigorous method of tensors and teams and their function in theoretical physics and utilized arithmetic. a specific target is to demystify tensors and supply a unified framework for knowing them within the context of classical and quantum physics. Connecting the part formalism conventional in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings should be a great addition to the literature on tensors and crew 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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Extra info for An Introduction to Tensors and Group Theory for Physicists

Example text

6 This problem will explore the notion of the Hermitian adjoint of a linear operator. (a) Let A be a linear operator on a finite-dimensional, real or complex vector space V with inner product (· | ·). Using the transpose AT from the previous problem, as well as the map L : V → V ∗ defined in Sect. 7, we can construct a new linear operator A† ; this is known as the Hermitian adjoint of A, and is defined as A† ≡ L−1 ◦ AT ◦ L : V → V . 44) which is the equation that is usually taken as the definition of the adjoint operator.

Given a Hilbert space H and an orthonormal (and possibly infinite) set {ei } ⊂ H, the set {ei } is said to be an orthonormal basis for H if (ei |f ) = 0 ∀i ⇒ f = 0. 14 below) that in the finite-dimensional case this definition is equivalent to our previous definition of an orthonormal basis. In the infinite-dimensional case, however, this definition differs substantially from the old one in that we no longer require Span{ei } = H (recall that spans only include finite linear combinations). Does this mean, though, that we now allow arbitrary infinite combinations of the basis vectors?

Also let f, h ∈ V ∗ . 22. 46) so that [g]−1 is truly a matrix representation of a non-degenerate bilinear form on V ∗ . 8 In this problem we will acquaint ourselves with P (R), the set of polynomials in one variable x with real coefficients. We will also meet several bases for this space which you should find familiar. 47) where n is arbitrary. Verify that P (R) is a (real) vector space. Then show that P (R) is infinite-dimensional by showing that, for any finite set S ⊂ P (R), there is a polynomial that is not in Span S.

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


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