Tensors: Significance and Features

सार

At the first face-off with Tensors, they appear to be scary and formidable. Even Einstein had to struggle a lot to master it. But once well versed with the subject Einstein used it aptly in the formulation of his masterpiece General Theory of Relativity (GTR), whose fame in due time made the subject of tensors synonymous with GTR in Physics. Earlier, however, with their little use in other areas of physics, students not opting for GTR could turn a blind eye to them. But since GTR much water has flown down the tensor pipeline, and the subject has evolved a lot with numerous applications not only in Physics and Mathematics but in fields as varied as Computer Science, Chemistry, Geology, Statistics, Medicine, Engineering, etc. In Physics, it is now regarded as an indispensable tool for the description of all the four fundamental interactions. Further, an operation on tensors called tensor product is a pre-requisite for the description of quantum states when two or more quantum systems get together, and also, the entangled states in their joint vector space need tensors for their expression. Very significantly, the fifth aspect of tensors, apart from its ability to represent invariance, anisotropicity, many quantum systems states and entanglement, is the capacity for large data storage, which is an artifact of the fact that high-rank tensors can be effectively represented by multi-dimensional hyper matrices. This feature of tensors has come to great advantage in Computer Science, where it is utilised for organising or storing large data and data mining with bearing on machine learning, deep learning, tensor imaging, face recognition, computer vision, etc. This article is a modest attempt (as the subject is deep and profound and cannot be justified in an article of over a dozen pages) to make accessible the features of tensors and their significance to the undergraduate students. The goal here is not to provide the students a working knowledge of tensors but to entice them by showing them the wonderful world of tensors so that they learn it on their own.