Tensors — again, again, again

“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. 

Then on pp. 7 and 8 he shows how a change of basis matrix (a 3 x 3 matrix written A^rs where rs, is one of 1, 2, 3) with nonZero determinant) gives the (usually incomprehensible) formula 

 

T^i’j’ = A^ik * A^jl T * (k, l)  where i, j, k, l are one of x, y, and z (or 1, 2, 3 as usually written)

 

So now you have a handle on the cryptic algebraic expression for tensors and what happens to them on a change of basis (e.g. how they transform).  Not bad for 5 pages of work — certainly not everything, but enough to make you comfortable with what follows — dual vectors, invariance, symmetric etc. etc.

 

Just knowing the multilinearity of tensors and just 2 postulates of quantum mechanics is all you need to understand entanglement — yes truly.  Yes, and you don’t need the Schrodinger equation, or differential equations at all, just linear algebra. 

 

Here is an old post to show you exactly how this works

 

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.

Tensors yet again

In the grad school course on abstract algebra I audited a decade or so ago, the instructor began the discussion about tensors by saying they were the hardest thing in mathematics. Unfortunately I had to drop this section of the course due a family illness. I’ve written about tensors before and their baffling notation and nomenclature. The following is yet another way to look at them which may help with their confusing terminology

First, this post will assume you have a significant familiarity with linear algebra. I’ve written a series of posts on the subject if you need a brush up — pretty basic — here’s a link to the first post — https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/
All of them can be found here — https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/.

Here’s another attempt to explain them — which will give you the background on dual vectors you’ll need for this post — https://luysii.wordpress.com/2015/06/15/the-many-ways-the-many-tensor-notations-can-confuse-you/

To the physicist, tensors really represent a philosophical position — e.g. there are shapes and processes external to us which are real, and independent of the way we choose to describe them mathematically. E. g. describing them by locating their various parts and physical extents in some sort of coordinate system. That approach is described here — https://luysii.wordpress.com/2014/12/08/tensors/

Zee in one of his books defines tensors as something that transforms like a tensor (honest to god). Neuenschwander in his book says “What kind of a definition is that supposed to be, that doesn’t tell you what it is that is changing.”

The following approach may help — it’s from an excellent book which I’ve not completely gotten through — “An Introduction to Tensors and Group Theory for Physicists” by Nadir Jeevanjee.

He says that tensors are just functions that take a bunch of vectors and return a number (either real or complex). It’s a good idea to keep the volume tensor (which takes 3 vectors and returns a real number) in mind while reading further. The tensor function just has one other constraint — it must be multilinear (https://en.wikipedia.org/wiki/Multilinear_map). Amazingly, it turns out that this is all you need.

Tensors are named by the number of vectors (written V) and dual vectors (written V*) they massage to produce the number. This is fairly weird when you think of it. We don’t name sin (x) by x because this wouldn’t distinguish it from the zillion other real valued functions of a single variable.

So an (r, s) tensor is named by the ordered array of its operands — (V, …V,V*, …,V*) with r V’s first and s V* next in the array. The array tells you what the tensor function must be.

How can Jeevanjee get away with this? Amazingly, multilinearity is all you need. Recall that the great thing about the linearity of any function or operator on a vector space is that ALL you need to know is what the function or operator does to the basis vectors of the space. The effect on ANY vector in the vector space then follows by linearity.

Going back to the volume tensor whose operand is (V, V, V) and the vector space for all 3 V’s (R^3), how many basis vectors are there for V x V x V ? There are 3 for each V meaning that there are 3^3 = 27 possible basis vectors. You probably remember the formula for the volume enclosed by 3 vectors (call them u, v, w). The 3 components of u are u1 u2 and u3.

The volume tensor calculates volume by ( U crossproduct V ) dot product W.
Writing the calculation out

Volume = u1*v2*w3 – u1*v3*w2 + u2*v3*w1 – u2*v1*w3 + u3*v1*w2 – u3*v2*w1. What about the other 21 combinations of basis vectors? They are all zero, but they are all present in the tensor.

While any tensor manipulating two vectors can be expressed as a square matrix, clearly the volume tensor with 27 components can not be. So don’t confuse tensors with matrices (as I did).

Note that the formula for volume implicitly used the usual standard orthogonal coordinates for R^3. What would it be in spherical coordinates? You’d have to use a change of basis matrix to (r, theta, phi). Actually you’d have to have 3 of them, as basis vectors in V x V x V are 3 places arrays. This gives the horrible subscript and superscript notation of matrices by which tensors are usually defined. So rather than memorizing how tensors transform you can derive things like

T_i’^j’ = (A^k_i’)*(A^k_i’) * T_k^l where _ before a letter means subscript and ^ before a letter means superscript and A^k_i’ and A^k_i’ are change of basis matrices and the Einstein summation convention is used. Note that the chance of basis formula for tensor components for the volume tensor would have 3 such matrices, not two as I’ve shown.

One further point. You can regard a dual vector as a function that takes a vector and returns a number — so a dual vector is a (1,0) tensor. Similarly you can regard vectors as functions that take dual vectors and returns a number, so they are (0,1) tensors. So, actually vectors and dual vectors are tensors as well.

The distinction between describing what a tensor does (e.g. its function) and what its operands actually are caused me endless confusion. You write a tensor operating on a dual vector as a (0, 1) tensor, but a dual vector is a (1,0) considered as a function.

None of this discussion applies to the tensor product, which is an entirely different (but similar) story.

Hopefully this helps

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.