13 - Tensors and tensor fields

Summary

Tensors and their space-time symmetries

Condensed matter systems have diverse physical properties that are described by tensorial quantities. By this we mean that physical properties of a system are usually defined by, and consist of, relationships between two or more particular measurable quantities associated with the system. These measurable quantities themselves usually assume tensorial forms, so that the ensuing physical properties that characterize a physical system are represented by tensors.

Scalars, i.e. tensors of rank 0, are typified by temperature, pressure, and mass of a homogeneous system, etc., while tensors of rank 1, i.e. vectorial properties, are manifest in electric and magnetic fields and moments, temperature gradients, and currents. Examples of second rank tensors are evident in the characterization of stress and strain in material systems. All these tensors describe either some physical state of a system or some externally applied physical field. For example, the magnitude and direction of the electric polarization is specified in response to an applied external electric field. We call all these tensors, both applied and induced, physical tensors.

We also encounter second and higher rank tensors that relate applied and induced physical tensors, for example, the electric susceptibility tensor of rank 2 connects the electric polarization with the applied electric fields, and the elasticity tensor, of rank 4, relates the strain to the applied stress. In contrast to physical tensors, the latter tensors are system specific: their particular forms, i.e. the number of linearly independent components and their values, are determined by the symmetry and structure of a given system. We refer to these system-specific tensors as matter, or material, tensors.