Abstract
This chapter 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 it 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. We then define the tensor product and uncover many applications of tensor products in classical and quantum physics. We close with a discussion of symmetric and anti-symmetric tensors, with examples concerning determinants and pseudovectors. The connection between anti-symmetric tensors and rotations is made, which leads naturally to the subject of Lie Groups and Lie Algebras in Part II.
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Notes
- 1.
The desire to raise and lower indices at will is one reason why we offset tensor indices and write T(e i , e j) as \(T_{i }^{\ j}\), rather than T i j. Raising and lowering indices on the latter would be ambiguous!
- 2.
Here and below we set all physical constants such as c and ε 0 equal to 1.
- 3.
This is also why we wrote the upper index directly above the lower index, rather than with a horizontal offset as is customary for tensors. For more about these numbers and a possible interpretation, see the beginning of the next section.
- 4.
This convention, in which we prime the indices rather than the objects themselves, is sometimes known as the Schouten convention; for more on this, see Battaglia and George [3].
- 5.
For a more detailed and complete discussion of covariance and contravariance, see Fleisch [5].
- 6.
See [8] for details.
- 7.
- 8.
We assume here that the basis vector e t satisfying \(\eta (e_{t},e_{t}) = -1\) is the fourth vector in the basis, which isn’t necessary but is somewhat conventional in physics.
- 9.
If the sleight-of-hand with the primed and unprimed indices in the last couple steps of (3.30) bothers you, puzzle it out and see if you can understand it. It may help to note that the prime on an index doesn’t change its numerical value; it’s just a reminder that it refers to the primed basis.
- 10.
For details on why the eigenvectors of Hermitian operators form a basis, at least in the finite-dimensional case, see Hoffman and Kunze [13].
- 11.
- 12.
We don’t bother here with index positions since most quantum mechanics texts don’t employ Einstein summation convention, preferring instead to explicitly indicate summation.
- 13.
Recall that we’ve set all physical constants such as c and ε 0 equal to 1.
- 14.
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, focusing instead on the tensorial properties.
- 15.
One needs the exterior derivative, a generalization of the curl, divergence and gradient operators from vector calculus. See Schutz [18] for a very readable account.
- 16.
This requires the so-called rigged Hilbert space; see Ballentine [2].
- 17.
Working with the momentum eigenfunctions e ipx instead doesn’t help; though these are legitimate functions, they still are not square-integrable since \(\int _{-\infty }^{\infty }\vert e^{ipx}\vert ^{2}\,dx = \infty \) !
- 18.
The Fourier transform of a function ψ(x) is often alternatively defined as
$$\displaystyle{ \phi (p) \equiv \int _{-\infty }^{\infty }dx\,\psi (x)e^{-ipx}, }$$which in Dirac notation would be written \(\langle p\vert \psi \rangle\). These two equivalent definitions of ϕ(p) are totally analogous to the two expressions (3.52) and (3.54) for ψ(x), and are again just the two interpretations of components discussed below (3.43).
- 19.
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’re 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 [15].
- 20.
\(L^{2}(\mathbb{R}^{3})\) is actually defined to be the set of all square-integrable functions on \(\mathbb{R}^{3}\), i.e. functions f satisfying
$$\displaystyle{ \int _{-\infty }^{\infty }\int _{ -\infty }^{\infty }\int _{ -\infty }^{\infty }dx\,dy\,dz\,\vert f\vert ^{2} < \infty. }$$Not too surprisingly, this space turns out to be identical to \(L^{2}(\mathbb{R}) \otimes L^{2}(\mathbb{R}) \otimes L^{2}(\mathbb{R})\).
- 21.
The number of transpositions required to get a given rearrangement is not unique, of course, but hopefully you can convince yourself that it’s 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’ll discuss this further in Chap. 4, specifically in Example 4.25.
- 22.
In relativistic quantum field theory, as opposed to non-relativistic quantum mechanics, this fact is no longer an additional postulate but rather an internally deducible fact, known as the spin-statistics theorem. The spin-statistics theorem furthermore states that bosons have integer spin and fermions have half-integral spin. See Zee [24] for a discussion and further references. It is also possible to deduce the symmetrization postulate from our assumption that n-particle states are invariant under permutation operators such as P from (3.69); proving this requires group theory, however, and so is postponed to Sect. 5.3
- 23.
A cyclic permutation of {1, …, n} is any rearrangement of {1, …, n} obtained by successively moving numbers from the beginning of the sequence to the end. That is, {2, …, n, 1}, { 3, …, n, 1, 2}, and so on are the cyclic permutations of {1, …, n}. Anti-cyclic permutations are cyclic permutations of {n, n − 1, …, 1}.
- 24.
- 25.
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 \(\mathbb{R}^{3}\), since even in \(\mathbb{R}^{2}\) a rotation can’t 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 \(\mathbb{R}^{3}\), see Goldstein [8].
- 26.
i.e., could not be made bigger.
- 27.
To map the components of e i ∧ e j to a matrix you’ll need the convention discussed in Box 2.4.
References
L. Ballentine, Quantum Mechanics: A Modern Development (World Scientific, Singapore, 1998)
F. Battaglia, T.F. George, Tensors: a guide for undergraduate students. Am. J. Phys. 81(7), 498–511 (2013)
D. Fleisch, A Student’s Guide to Vectors and Tensors (Cambridge University Press, Cambridge, 2011)
S. Gasiorowicz, Quantum Physics, 2nd edn. (Wiley, New York, 1996)
H. Goldstein, Classical Mechanics, 2nd edn. (Addison-Wesley, Massachusetts, 1980)
K. Hoffman, D. Kunze, Linear Algebra, 2nd edn. (Prentice Hall, Englewood Cliffs, 1971)
M. Reed, B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic, San Diego, 1972)
J.J. Sakurai, Modern Quantum Mechanics, 2nd edn. (Addison Wesley Longman, Reading, MA, 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, New York, 1979)
A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, Princeton, 2010)
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Jeevanjee, N. (2015). Tensors. In: An Introduction to Tensors and Group Theory for Physicists. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-14794-9_3
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