“A tensor is something that transforms like a tensor” — and a duck is something that quacks like a duck. If you find this sort of thing less than illuminating, I’ve got the book for you — “An Introduction to Tensors and Group Theory for Physicists” by Nadir Jeevanjee.
He notes that many physics books trying to teach tensors start this way, without telling you what a tensor actually is.
Not so Jeevanjee — right on the first page of text (p. 3) he says “a tensor is a function which eats a certain number of vectors (known as the rank r of the tensor) and produces a number. He doesn’t say what that number is, but later we are told that it is either C or R.
Then comes the crucial fact that tensors are multilinear functions. From that all else flows (and quickly).
This means that you know everything you need to know about a tensor if you know what it does to its basis vectors.
He could be a little faster about what these basis vectors actually are, but on p. 7 you are given an example explicitly showing them.
To keep things (relatively) simple the vector space is good old 3 dimensional space with basis vectors x, y and z.
His rank 2 tensor takes two vectors from this space (u and v) and produces a number. There are 9 basis vectors not 6 as you might think — x®x, x®y, x®z, y®x, y®y, y®z, z®x, z®y, and z®z. ® should be read as x inside a circle
Tensor components are the (real) numbers the tensor assigns to the 9 — these are written T(x®x) , T(x®y) T( x®z), T(y®x), T(y®y), T(y®z), T(z®x), T(z®y), and T(z®z)– note that there is no reason that T(x®y) should equal T(y®x) any more than a function R^2 –> R should give the same values for (1, 2) and (2, 1).
One more complication — where do the components of u and v fit in? u is really (u^1, u^2, u^3) and v is really (v^1, v^2, v^3)
They multiply each other and the T’s — so the first term of the tensor (sometimes confusingly called a tensor component)
is u^1 * v^1 * T(x®x) and the last is u^3 * v^3 T(z®z). Then the 9 tensor terms/components are summed giving a number.
How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement
Tensors continue to amaze. I never thought I’d get a simple mathematical explanation of entanglement, but here it is. Explanation is probably too strong a word, because it relies on the postulates of quantum mechanics, which are extremely simple but which lead to extremely bizarre consequences (such as entanglement). As Feynman famously said ‘no one understands quantum mechanics’. Despite that it’s never made a prediction not confirmed by experiments, so the theory is correct even if we don’t understand ‘how it can be like that’. 100 years of correct prediction of experimentation are not to be sneezed at.
If you’re a bit foggy on just what entanglement is — have a look at https://luysii.wordpress.com/2010/12/13/bells-inequality-entanglement-and-the-demise-of-local-reality-i/. Even better; read the book by Zeilinger referred to in the link (if you have the time).
Actually you don’t even need all the postulates for quantum mechanics (as given in the book “Quantum Computation and Quantum Information by Nielsen and Chuang). No differential equations. No Schrodinger equation. No operators. No eigenvalues. What could be nicer for those thirsting for knowledge? Such a deal ! ! ! Just 2 postulates and a little formal mathematics.
Postulate #1 “Associated to any isolated physical system, is a complex vector space with inner product (that is a Hilbert space) known as the state space of the system. The system is completely described by its state vector which is a unit vector in the system’s state space”. If this is unsatisfying, see an explication of this on p. 80 of Nielson and Chuang (where the postulate appears)
Because the linear algebra underlying quantum mechanics seemed to be largely ignored in the course I audited, I wrote a series of posts called Linear Algebra Survival Guide for Quantum Mechanics. The first should be all you need. https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ but there are several more.
Even though I wrote a post on tensors, showing how they were a way of describing an object independently of the coordinates used to describe it, I did’t even discuss another aspect of tensors — multi linearity — which is crucial here. The post itself can be viewed at https://luysii.wordpress.com/2014/12/08/tensors/
Start by thinking of a simple tensor as a vector in a vector space. The tensor product is just a way of combining vectors in vector spaces to get another (and larger) vector space. So the tensor product isn’t a product in the sense that multiplication of two objects (real numbers, complex numbers, square matrices) produces another object of the exactly same kind.
So mathematicians use a special symbol for the tensor product — a circle with an x inside. I’m going to use something similar ‘®’ because I can’t figure out how to produce the actual symbol. So let V and W be the quantum mechanical state spaces of two systems.
Their tensor product is just V ® W. Mathematicians can define things any way they want. A crucial aspect of the tensor product is that is multilinear. So if v and v’ are elements of V, then v + v’ is also an element of V (because two vectors in a given vector space can always be added). Similarly w + w’ is an element of W if w an w’ are. Adding to the confusion trying to learn this stuff is the fact that all vectors are themselves tensors.
Multilinearity of the tensor product is what you’d think
(v + v’) ® (w + w’) = v ® (w + w’ ) + v’ ® (w + w’)
= v ® w + v ® w’ + v’ ® w + v’ ® w’
You get all 4 tensor products in this case.
This brings us to Postulate #2 (actually #4 on the book on p. 94 — we don’t need the other two — I told you this was fairly simple)
Postulate #2 “The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.”
http://planetmath.org/simpletensor
Where does entanglement come in? Patience, we’re nearly done. One now must distinguish simple and non-simple tensors. Each of the 4 tensors products in the sum on the last line is simple being the tensor product of two vectors.
What about v ® w’ + v’ ® w ?? It isn’t simple because there is no way to get this by itself as simple_tensor1 ® simple_tensor2 So it’s called a compound tensor. (v + v’) ® (w + w’) is a simple tensor because v + v’ is just another single element of V (call it v”) and w + w’ is just another single element of W (call it w”).
So the tensor product of (v + v’) ® (w + w’) — the elements of the two state spaces can be understood as though V has state v” and W has state w”.
v ® w’ + v’ ® w can’t be understood this way. The full system can’t be understood by considering V and W in isolation, e.g. the two subsystems V and W are ENTANGLED.
Yup, that’s all there is to entanglement (mathematically at least). The paradoxes entanglement including Einstein’s ‘creepy action at a distance’ are left for you to explore — again Zeilinger’s book is a great source.
But how can it be like that you ask? Feynman said not to start thinking these thoughts, and if he didn’t know you expect a retired neurologist to tell you? Please.
Chemiotics II 