Abstract
Chapter 3 begins with the abstract, coordinate-free definition of a tensor. This definition is standard in the math literature and in texts on General Relativity, but is otherwise not accessible in the physics literature. A major feature of this book is that is provides a relatively quick route to this definition, without the full machinery of differential geometry and tensor analysis. After the definition and some examples we thoroughly discuss change of bases and make contact with the usual coordinate-dependent definition of tensors. Matrix equations for a change of basis are also given. This is followed by a discussion of active and passive transformations, a subtle topic that is rarely fleshed out fully in other texts. We then define the tensor product and uncover many applications of tensor products in classical and quantum physics; in particular, we discuss the unwritten rule that adding degrees of freedom in Quantum Mechanics means taking the tensor product of the corresponding Hilbert spaces, and we give several examples. We then close with a discussion of symmetric and antisymmetric tensors. Important machinery such as the wedge product is introduced, along with examples concerning determinants and pseudovectors. The connection between antisymmetric tensors and rotations is made, which leads naturally to the subject of Lie Groups and Lie Algebras in Part II.
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- 1.
Here and below we set all physical constants such as c and ϵ 0 equal to 1.
- 2.
This is also why we wrote the upper index directly above the lower index, rather than with a horizontal displacement as is customary for tensors. For more about these numbers and a possible interpretation, see the beginning of the next section.
- 3.
See Problem 3.1 for more on orthogonal matrices, as well as Chap. 4.
- 4.
We assume here that the basis vector e t satisfying η(e t ,e t )=−1 is the fourth vector in the basis, which is not necessary but is somewhat conventional in physics.
- 5.
If the sleight-of-hand with the primed and unprimed indices in the last couple steps of (3.38) bothers you, puzzle it out and see if you can understand it. It may help to note that the prime on an index does not change its numerical value, it is just a reminder that it refers to the primed basis.
- 6.
For details on why the eigenvectors of Hermitian operators form a basis, at least in the finite-dimensional case, see Hoffman and Kunze [10].
- 7.
We do not bother here with index positions since most quantum mechanics texts do not employ Einstein summation convention, preferring instead to explicitly indicate summation.
- 8.
Recall that we have set all physical constants such as c and ϵ 0 equal to 1.
- 9.
In this example and the one above we are actually not dealing with tensors but with tensor fields, i.e. tensor-valued functions on space and spacetime. For the discussion here, however, we will ignore the spatial dependence of the fields, focusing instead on the tensorial properties.
- 10.
One needs the exterior derivative, a generalization of the curl, divergence and gradient operators from vector calculus. See Schutz [15] for a very readable account.
- 11.
This requires the so-called ‘rigged’ Hilbert space; see Ballentine [2].
- 12.
Working with the momentum eigenfunctions e ipx instead does not help; though these are legitimate functions, they still are not square-integrable since \(\int_{-\infty}^{\infty}|e^{ipx}|^{2} \,dx=\infty\)!
- 13.
You may have noticed that we defined tensor products only for finite-dimensional spaces. The definition can be extended to cover infinite-dimensional Hilbert spaces, but the extra technicalities needed do not add any insight to what we are trying to do here, so we omit them. The theory of infinite-dimensional Hilbert spaces falls under the rubric of functional analysis, and details can be found, for example, in Reed and Simon [12].
- 14.
L 2(ℝ3) is actually defined to be the set of all square-integrable functions on ℝ3, i.e. functions f satisfying
Not too surprisingly, this space turns out to be identical to L 2(ℝ)⊗L 2(ℝ)⊗L 2(ℝ).
- 15.
The number of transpositions required to get a given rearrangement is not unique, of course, but hopefully you can convince your self that it is always odd or always even. A rearrangement which always decomposes into an odd number of transpositions is an odd rearrangement, and even rearrangements are defined similarly. We will discuss this further in Chap. 4, specifically in Example 4.22.
- 16.
For a complete treatment, however, you should consult Hoffman and Kunze [10], Chap. 5.
- 17.
No doubt you are used to thinking about a rotation as a transformation that preserves distances and fixes a line in space (the axis of rotation). This definition of a rotation is particular to ℝ3, since even in ℝ2 a rotation cannot be considered to be “about an axis” since \(\hat{\mathbf{z}}\notin{\mathbb{R}}^{2}\). For the equivalence of our general definition and the more intuitive definition in ℝ3, see Goldstein [6].
- 18.
I.e. could not be made bigger.
References
L. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, Singapore, 1998
S. Gasiorowicz, Quantum Physics, 2nd ed., Wiley, New York, 1996
H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, 1980
K. Hoffman and D. Kunze, Linear Algebra, 2nd ed., Prentice Hall, New York, 1971
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, San Diego, 1972
J.J. Sakurai, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, Reading, 1994
B. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, Cambridge, 1980
S. Sternberg, Group Theory and Physics, Princeton University Press, Princeton, 1994
F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, Berlin, 1979
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Jeevanjee, N. (2011). Tensors. In: An Introduction to Tensors and Group Theory for Physicists. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4715-5_3
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