## Tag Archives: Tensor product

### The many ways the many tensor notations can confuse you

This post is for the hardy autodictats attempting to learn tensors on their own. If you use multiple sources, you’ll find that they define the same terms used to describe tensors in diametrically opposed ways, so that just when you thought you knew what terms like covariant and contravariant tensor meant,  another source defines them completely differently, leading you to wonder (1) about your intelligence (2) your sanity.

Tensors involve vector spaces and their bases. This post assumes you know what they are. If you don’t understand how a vector can be expressed in terms of coordinates relative to a basis, pick up any book on linear algebra.

Tensors can be defined by the way their elements transform under a change of coordinate basis. This is where the terms covariant and contravariant come from. By the way when Einstein says that physical quantities must transform covariantly, he means they transform like tensors do (even contravariant tensors).

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Addendum 12/17 — Neuenschwander “Tensor Calculus for Physicists 2015 pp. 17 –> There are 3 meanings to the term covariance

l. Covariance of an equation — it transforms under a coordinate transformation the way the coodinates do.

2. A way of distinguishing between covariant and contravariant vectors (see below).

3. A type of tensor derivative — a modification of the usual derivative definition to make the derivative of a tensor another tensor (the usual derivative definition fails this).

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True enough, but this approach doesn’t help you understand the term tensor product or the weird ® notation (where there is an x within the circle) used to describe it.

The best way to view tensors (from a notational point of view) is to look on them as functions which take finite Cartesian products (https://en.wikipedia.org/wiki/Cartesian_product) of vectors and covectors and produce a single real number.

To understand what a covector (aka dual vector) is, you must understand the inner product (aka dot product).

The definition of inner product (dot product) of a vector V with itself written , probably came from the notion of vector length. Given the standard basis in two dimensional space E1 = (1,0) and E2 = (0,1) all vectors V can be written as x * E1 + y * E2 (x is known as the coefficient of E1). Vector length is given by the good old Pythagorean theorem as SQRT[ x^2 + y^2]. The dot product (inner product) is just x^2 + y^2 without the square root.

In 3 dimensions the distance of a point (x, y, z) from the origin is SQRT [x^2 + y^2 + z^2]. The definition of vector length (or distance) easily extends (by analogy) to n dimensions where the length of V is SQRT[x1^2 + x2^2 + . . . . + xn^2] and the dot product is x1^2 + x2^2 + . . . . + xn^2. Length is always a non-negative real number.

The definition of inner product also extends to the the dot product of two different vectors V and W where V = v1 * E1 + v2 * E2 + . … vn * En, W = w1 * E1 + . . + wn * En — e.g.  = v1 * w1 + v2 * w2 + . . . + vn * wn. Again always a real number, but not always positive as any of the v’s and w’s can be negative.

So, if you hold W constant you can regard it as a function on the vector space in which V and W reside which takes any V and produces a real number. You can regard V the same way if you hold it constant.

Now with some of the complications which mathematicians love, you can regard the set of functions { W } operating on a vector space, as a vector space itself. Functions can be added (by their results) and can be multiplied by a real number (a scalar). The set of functions { W } regarded as a vector space is called the dual vector space.

Well if { W } along with function addition and scalar multiplication is a vector space, it must have a basis. Everything I’ve every read about tensors  involves finite dimensional vector spaces. So assume the vector space A is n dimensional where n is an positive integer, and call its basis vectors the ordered set a1, . . . , an. The dual vector space (call it B) is also n dimensional with another basis the ordered set b1, . . . , bn.

The bi are chosen so that their dot product with elements of A’s basis = Kronecker delta, e.g. if i = j then
= 1. If i doesn’t equal j  then  = 0. This can be done by a long and horrible process (back in the day before computer algebra systems) called Gram Schmidt orthonormalization. Assume this can be done. If you’re a true masochist have a look at https://en.wikipedia.org/wiki/Gram–Schmidt_process.

Notice what we have here. Any particular element of the dual space B (a real valued function operating on A) call it f can be written down as f1 * b1 + . . . + fn * bn. It will take any vector in A (written g1 * a1 + . . . + gn * an) and give you f1 * g1 + . . . + fn * gn which is a real number. Basically any element ( say bj) of the basis of dual space B just looks at a vector in A and picks out the coefficient of aj (when it forms the dot product with the vector in A.

Now (at long last) we can begin to look at the contrary way tensors are described. The most fruitful way is to look at them as the product of individual dot products between a vector and a dual vector.

Have a look at — https://luysii.wordpress.com/2014/12/08/tensors/. To summarize  — the whole point of tensor use in physics is that they describe physical quantities which are ‘out there’ independently of the coordinates used to describe them. A hot dog has a certain length independently of its description in inches or centimeters. Change your viewpoint and the its coordinates in space will change as well (the hot dog doesn’t care about this). Tensors are a way to accomplish this.

It’s too good to pass up, but the length of the hot dog stays the same no matter how many times you (non invasively) measure it.  This is completely different than the situations in quantum mechanics, and is one of the reasons that quantum mechanics has never been unified with general relativity (which is a theory of gravity based on tensors).

Remember the dot product concerns   . If you change the basis of vector  W (so vector W has different coordinates) the basis of dual vector   V must also change (to keep the dot product the same). A choice must be made as to which of the two concurrent basis changes is fundamental (actually neither is as they both are).

Mathematics has chosen the basis of vector W in as fundamental.

When you change the basis of W, the coefficients of W must change in the opposite way (to keep the vector length constant). The coefficients of W are said to change contravariantly. What about the coefficients of V? The basis of V changes oppositely to the basis of W (e.g. contravariantly), so the coefficients of V must change differently from this e.g. the same way the basis of W changes — e.g. covariantly. Confused?  Nonetheless, that’s the way they are named

Vectors and convectors and other mathematical entities such differentials, metrics and gradients are labelled as covariant or contravariant by the way their numerical coefficients change with a change in basis.

So the coefficients of vector W transform contravariantly, and the coefficients of dual vector V transform covariantly. This is true even though the coefficients of V and W always transform contravariantly (e. g. oppositely) to the way their basis transforms.

An immense source of confusion.

As mentioned above, one can regard vectors and dual vectors as real valued functions on elements of a vector space. So (adding to the confusion) vectors and dual vectors are both tensors. Vectors are contravariant tensors, and dual vectors are covariant tensors.

Now we form Cartesian products of vectors W (now called V) and convectors V (hereafter called V* to keep them straight).

We get something like this V x V x V x V* x V*, a cartesian product of 3 contravariant vectors and 2 dual vectors.

To get a real number out of them we form the tensor product V* ® V* ® V* ® V ® V, where the first V* operates on the first V to produce a real number, the second operates . . . and the last V* operates on the last V to produce a real number. All real numbers produced are multiplied together to produce the result.

Why not just call  V* ® V* ® V* ® V ® V a product? Well each V and V* is an n dimensional vector space, and the tensor V ® V is a n^2 dimensional space (and  V* ® V* ® V* ® V ® V is an n^5 dimensional vector space). When we form the product of two numbers (real or complex) we just get another number of the same species (real or complex). The tensor product of two n dimensional vector spaces is not another n dimensional space, hence the need for the adjective modifying the name product. The dot product nomenclature is much the same, the dot product of two vectors is not another vector, but a real number.

Here is yet another source of confusion. What we really have is a tensor product V* ® V* ® V* ® V ® V operating on a Cartesian product of vectors and covectors (tensors themselves) V x V x V x V* x V* to produce a real number.

Tensors can either be named by their operands making this a 3 contravariant 2 covariant tensor — (3, 2) tensor.

Other books name them by their operator (e.g. the tensor product) making it a 3 covariant 3 contravariant tensor (a 2, 3) tensor.

If you don’t get this settled when you switch books you’ll think you don’t really understand what contravariant and covariant mean (when in fact you do). Mercifully, one constancy in notation (thankfully) is that the contravariant number always comes first (or on top) and the covariant number second (or on bottom).

Hopefully this is helpful.  I wish I’d had this spelled out when I started.

### 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.