# Tensors

## 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.

## Keywords

Tensor Product Basis Vector Symmetric Tensor Vector Space Versus Antisymmetric Tensor## References

- 2.L. Ballentine,
*Quantum Mechanics: A Modern Development*, World Scientific, Singapore, 1998 Google Scholar - 5.S. Gasiorowicz,
*Quantum Physics*, 2nd ed., Wiley, New York, 1996 Google Scholar - 6.H. Goldstein,
*Classical Mechanics*, 2nd ed., Addison-Wesley, Reading, 1980 Google Scholar - 10.K. Hoffman and D. Kunze,
*Linear Algebra*, 2nd ed., Prentice Hall, New York, 1971 Google Scholar - 12.M. Reed and B. Simon,
*Methods of Modern Mathematical Physics I: Functional Analysis*, Academic Press, San Diego, 1972 Google Scholar - 14.J.J. Sakurai,
*Modern Quantum Mechanics*, 2nd ed., Addison-Wesley, Reading, 1994 Google Scholar - 15.B. Schutz,
*Geometrical Methods of Mathematical Physics*, Cambridge University Press, Cambridge, 1980 Google Scholar - 16.S. Sternberg,
*Group Theory and Physics*, Princeton University Press, Princeton, 1994 Google Scholar - 18.F. Warner,
*Foundations of Differentiable Manifolds and Lie Groups*, Springer, Berlin, 1979 Google Scholar