Mathematical rigor in theoretical physics

We have seen in the last post that mathematical sloppiness can easily lead you astray. But is this always the case?

This is not a new problem. John von Neumann sorted out the mathematical foundation of quantum mechanics in Mathematical Foundations of Quantum Mechanics. The first time I read this book I found it incredibly boring: this is what you learn in school. Then I learned to appreciate its sheer brilliance. The reason it looks so boring is because it was so good it became the standard. At the same time, competing with von Neumann was Dirac who introduced the well known “Dirac functions” – an invaluable tool for any quantum mechanics computation. Here is what von Neumann had to say about it:

“The method of Dirac [...] in no way satisfies the requirements of mathematical rigor – not even if these are reduced in a natural and proper fashion to the extent common elsewhere in theoretical physics” – OUCH!!!

Now I am not a historian and I don’t know the year von Neumann wrote the book (I have only the translation to English year), but it was probably in the 1930s well before the theory of distributions put Dirac’s delta function on a solid mathematical foundation.

Fast forward to present, I have found a series of outstanding lectures by Carl Bender which shook me to the core regarding to what it means to be a theoretical physicist. Towards the end of the series, the fog clears and I came back to my original beliefs about mathematical rigor along the lines of von Neumann, but Mr. Bender managed to give me a scare with some mathematical voodoo.

To give you a taste of the lectures, let me ask a question from Lecture 4: How much is:

1 - 1 + 1 - 1 + … = ?

This is stupid you may say: it is clearly a divergent series. Worse you can make it converge to any number. You pick one, say 26, Add the first 26 ones and then cancel the rest of the series. Does Hilbert hotel ring a bell?

Now how about this series:

1 + 0 – 1 + 1 + 0 – 1 + 1 + 0 – 1 + … = ?

Would it surprise you if I can prove that this gives a different answer than the first series? And all that we have extras are an infinite numbers of zeros!!!

Let’s proceed.

First we can introduce the Euler summation machine which takes a divergent series and spits out a number E:

So let our sum: Sum (a_n) be not convergent. Construct the following function:

f(x) = Sum(a_n x^n) for x < 1 where x is such that the sum converges

Define E=lim_{x->1} f(x)

Let’s apply it to: 1 – 1 + 1 – 1 + 1 – 1 …

f(x) = 1-x+x^2-x^3+… = 1/(1+x)

Therefore E = 1/2

Can we make other machines in this spirit?

Yes, and here is another one, the Borel summation:

Again Sum (a_n) is not convergent.

We know that: Integral dt exp^(-t) t^n = n! which means that

1 = Integral dt e^{-t} t^n / n!

Replace Sum (a_n) -> Sum (a_n)*1 = Sum (a_n)* Integral dt e^{-t} t^n / n!

Then flip the sum with the integral:

B = Integral dt exp^(-t) Sum (t^n a_ /n!)

Do E=B? Yes they do and here is why:

E and B are machines obeying two rules:

Rule 1: summation property

S(a0 + a1 + a2+ …) = a0 + S(a1 + a2+ …)

Rule 2: linearity

S( Sum(alpha a_n + beta b_n)) = alpha S( Sum (alpha a_n)) + beta S( Sum(b_n))

Let’s apply it to our two divergent series:

sum(1 -1 + 1 -1 + …) = S

S=1+ sum(-1 + 1 -1 + …) (by Rule 1)

S = 1 – sum(1 - 1 + 1 - 1 + …) (by Rule 2)

S = 1-S

2*S= 1

S= 1/2 BINGO!

Now the second series

S=           sum( 1 + 0 -1 +1 +0 -1 +1+…) =

S = 1+     sum( 0 – 1 +1+0 -1  +1 +0+…)=

S = 1+0+ sum(-1+ 1 +0 -1 +1  +0 -1+…)

3S = 1+1+0 +nothing(cancel term by term, no commutation of the order of the numbers in the series)

S = 2/3

Let’s double check with Euler:

f(x) = 1 – x^2 + x^3 – x^5 + x^6 –x^8 +…

= (1+x^3+x^6+…) – (x^2 + x^5 +x^8+…)

=1/(1-x^3) – x^2/(a-x^3) = (1-x^2)/(1-x^3)

lim x-> 1 f(1) = 2/3

Mr. Bender is also making provocative (but true) statements like:

“If you are given a series and you have to add it up the dumbest thing that you can possibly do is add it up […] and if the series diverges it’s not only a stupid idea, it doesn't work.”

Here is the complete series on You Tube:

Lecture 1:

http://www.youtube.com/watch?v=LYNOGk3ZjFM

Lecture 2:

http://www.youtube.com/watch?v=tV-xIhP7VU8

Lecture 3:

http://www.youtube.com/watch?v=_Sm7SNlNUOI

Lecture 4:

http://www.youtube.com/watch?v=VvqeJkT3uyo

Lecture 5:

http://www.youtube.com/watch?v=LMw0NZDM5B4

Lecture 6:

http://www.youtube.com/watch?v=nWKYrpKn3-0

Lecture 7:

http://www.youtube.com/watch?v=pyEQXMKfEnA

Lecture 8:

http://www.youtube.com/watch?v=YGDTW7mNUKM

Lecture 9:

http://www.youtube.com/watch?v=rs2iJugn808

Lecture 10:

http://www.youtube.com/watch?v=vem9Ltth_Hg

Lecture 11:

http://www.youtube.com/watch?v=9OO-gqj3N0c

Lecture 12:

http://www.youtube.com/watch?v=clpHq5Rs8a4

Lecture 13:

http://www.youtube.com/watch?v=ibR4k-HvRm4

Lecture 14:

http://www.youtube.com/watch?v=31TegmZpIyc

Lecture 15:

http://www.youtube.com/watch?v=RqZ0VsEc4F0

Enjoy!

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Holonomy in quantum mechanics

Bohm-Aharonov effect

Let’s start with the definition of holonomy: if you walk in  closed loop and the object you carry changes when you complete the loop then you have experienced a holonomy.

Now this sounds plain crazy so a simple example can illustrate it. Suppose you are a hunter living on the Equator and you go on a quest to explore the Earth. You walk a quarter of the Earth circumference on the Equator going east, you travel north all the way to the North Pole and then you go straight down to the starting point of your journey. During your journey you carry with you your spear always making sure it is pointing in the same direction. For definiteness sake, let’s say that originally your spear was pointing towards the North Pole. When you walk on the Equator and then towards the North Pole your spear is pointing north. However, on the last part of the journey, your spear will be pointing west. So upon your arrival the spear has a different orientation even though you always carried it parallel with itself. This is the result of Earth’s curvature.

Now something very similar happens in Einstein’s general relativity: the presence of mass curves the space-time and although you travel on geodesics in a straight line, nearby curves are not parallel. We feel this lack of parallelism as gravity.

Fine, we understand this, but what does holonomy have to do with quantum mechanics? Suppose I have a box with a quantum device which when I press a button can flash either a red light or a blue light. Suppose that every time I press the button only the red light will flash. Now I go on a similar quest on a closed space loop and when I press the button only the blue light is flashing. This would happen every time I would circle a zone of magnetic field although I cannot detect any magnetic field anywhere on my path.

Now this is downright freaky: there are no forces whatsoever along my path and still there is a measurable effect. Welcome to the wonderful Bohm Aharonov effect.

Mathematicians usually refers to geometric phases and call this effect a topological one. But surprisingly it has a nice mathematical explanation in terms of boundary conditions and domains in standard quantum mechanics. When learning quantum mechanics, pesky boundary conditions and domains tend to be ignored as pedantic crossing of t’s and dotting the i’s. This is the typical cavalier physicist’s attitude towards mathematical rigor. Don’t believe me? Ask any physicist to tell you the difference between Hermitean and self-adjoint. If one in one hundred knows the difference you are lucky.

The point is that self-adjointness demands the domains of the operator and its hermitean counterpart (complex conjugate and transposed) to be identical!!!

A hermitean operator may have even an infinity of different self-adjoint extensions. And the eigenvalues (the observed values) are all distinct.

Here is an example of how sloppiness can get you into trouble: http://www.mth.kcl.ac.uk/~streater/lostcauses.html#VIII

Now for the Bohm-Aharonov effect, it is the boundary condition which selects the eigenvalues. And this boundary condition comes from the magnetic flux carried by the solenoid.

An excellent review of those topics can be found in Asher Peres’ classic: “Quantum Theory: Concepts and Methods” and I hope I wetted your appetite to read this outstanding book.

Now, although true, changes without forces bother a lot a people. Last year a proposal was made to restore the role of forces in the Bohm-Aharonov effect: http://arxiv.org/abs/1110.6169 The paper has a clever idea: the apparatus and not the particle feels the force, but it is fundamentally flawed because any forces will give out the “which way information” which will destroy the interference pattern. Holonomy is a fundamental property of Nature and it is not explainable away.

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What is the number system of quantum mechanics?

Gauge theory and quantions

Let me start with the answer for the problem I posted last time. The hard part for finding the projectors in quantum mechanics when the number system is non-commutative is to realize the proper place to put the scalars:

|psi>lambda<psi|

with lambda*lambda = lambda. If the number system is a division algebra (like in the case of quaternionic quantum mechanics) by dividing with lambda it follows that lambda = 1. However this is not the case for quantionic quantum mechanics. Expressing the quantions as a linear combination of the Pauli matrices and the identity operator over complex numbers:

lambda = a0 I + a1 Sigma1 +a2 Sigma2 + a3 Sigma3

and using the algebraic properties of the Pauli matrices, from lambda*lambda = lambda one gets:

a0 = ½

a1^2 + a2^2+a3^2 = ½

which is the equation of a sphere. A sphere can be parameterized by two numbers: latitude and longitude. However in this case the latitude and longitude are complex numbers, and the parametrization needs 2X2 = 4 real numbers. This is where the overall coefficient of 4 is coming from.

Coming back to gauge theory, the gauge degrees of freedom leave invariant every observable in the physical system. So why bothering with them one may ask? Because they have very big physical consequences.

Let’s explain it using a nice analogy from everyday life. Think of a thick rug.

At each point in the rug there is a piece fiber sticking out. Suppose somebody drags something across the rug (it could be a toy track for example) and leaves a mark on it. How can we describe this local disturbance in rug’s fibers? If we look very closely we notice displacements between nearby fibers. Mathematicians call this a “connection” which allows to quantify correctly the notion of change (covariant derivative) and a notion of moving from one place to another (parallel transport). What this means is that if the rug is rolled up or twisted in some way, we need to add the local disturbance to the global rug twist to predict correctly the location of each fiber.

Fine, it is not hard to grasp those mathematical concepts but what this has to do with physics?

The remarkable fact is that what mathematicians call connections, physicists call potentials (like the electromagnetic potential).

It took some time to recognize the gauge theory mathematical structure in the electromagnetic field, but if you recall from Maxwell’s equations, there is an electromagnetic four potential A^µ defined up to a gauge and the electromagnetic tensor is F ^µν = ∂^µ A^ν - ∂^ν A^µ The covariant derivative in this case is D_µ = ∂_µ - i A_µ

Grgin’s book has a nice parallel between the gauge theory of electromagnetism, gravity, and quantionic quantum mechanics (electroweak gauge theory).

But where is the gauge degree of freedom coming from? It comes from the inner product in quantum mechanics over an arbitrary number system.  Basically it is the exponent part in a generalized polar form decomposition of the number system similar with complex numbers polar form decomposition.

So here is how it works: in each point of space-time we attach a “fiber” in the form of a quantion. If in ordinary quantum mechanics one has functions of complex variables, here we have functions of quantionic variables. A key difference is in normalization. In a quantionic quantum field, by Zovko interpretation, we demand that the conservation of a quantionic current stemming from the inner product  q^* q. Everything else follows from this.

In gauge theories, the “marks of the toy truck on the rug” are actually particles (lines in a Feynman diagram). The forces are generated by gradients of the potentials, and the potentials are the “connections” allowing the “parallel transport”, or the means to compare nearby points in space-time. In the Standard Model, there are three gauge symmetries: U(1), SU(2), SU(3) corresponding to three fundamental forces: electromagnetism, weak, and strong force. Grand unification theories (GUT) seek to find a common “underlying rug”. A general feature of GUTs is the cross talk between related “fibers” which means that particles are not stable and in particular the proton is not stable and is eventually decaying into leptons.

When a quantion is expressed as a 4x4 matrix, the null entries (8 of them) are used by the electroweak potentials. The covariant derivative is a right quantion and the commutativity between left quantions and right quantions assures the Leibnitz identity needed to turn a right quantion into a derivation.

In quantionic quantum mechanics there is also a notion of curvature and holonomy as well.

Quantionic quantum mechanics has this dual interpretation as ordinary quantum mechanics or as gauge theory. As ordinary quantum mechanics the distinct number system predicts a new physical phenomenon not present in complex quantum mechanics: the zitter effect.

Can ordinary complex quantum mechanics predict holonomy effects too? The answer is yes and we will cover it next time.

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What is the number system of quantum mechanics?

Inherently relativistic quantum mechanics

It was a cold 2007 January morning during rush our in L’enfant Plaza metro station in Washington DC when one of the best violinist virtuoso in the world, Joshua Bell

played on a Stradivarius violin some of the best classical pieces for about an hour. Do you think his masterpiece performance drew a crowd? The lottery kiosk nearby was attracting much more attention.

In the same 2007 year, Emile Grgin published his Structural Unification of Quantum Mechanics and Relativity book. Still to this day people buy into the “lottery” idea that quantum mechanics is solely about information and Born’s rule.

But maybe Grgin’s results were “crackpot”. That was the reaction of the archive when http://arxiv.org/abs/1204.3562 was reclassified from the quantum section to the general section dedicated to “laymen’s fantasies”.

So let’s prove quantionic quantum mechanics is the real deal and not some crazy idea. First quantions are the simplest non-trivial type I von Neumann algebra. All linear algebras have matrix and vector representations which come in pairs: left algebra and a column vector, and right algebra and a row vector.

For example, a complex number z = a + ib can be represented as:

a   -b

b  a

and

a

b

OR

a  b

-b a

and

a  b

It is a simple theorem that for linear associative algebras the left and right matrix representations commute. For quantions, the left and column representations are:

q1   q3   0    0

q2   q4   0    0

0     0    q1  q3

0     0    q2  q4

and

q1

q2

q3

q4

and the right and row representations are:

q1   0    q3   0

0    q1    0    q3

q2   0    q4   0

0    q2    0   q4

and

q1 q2 q3 q4

Both representations play a major role in the physics. For now, since in the 4x4 matrix algebra representation the left L and a right R quantions have non-overlapping parts, the von-Neumann double commutant theorem (http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem) holds: L^{``} = L In turn, this puts the corresponding quantionic Hilbert module theory on a solid foundation.

To bring the discussion on a more known ground, here is the 1-to-1 mapping between quantions and spinors:

q1                                 -Psi2

q2        <----->sqrt(2)     Psi3^*

q3                                   Psi1

q4                                   Psi4^*

Psi1                                 q3

Psi2    <----->1/sqrt(2)   -q1

Psi3                                 q2^*

Psi4                                 q4^*

Dirac’s current

j^mu = Psi^{dagger} gamma^0 gamma^mu Psi

is the same as:

j^mu = (q^* q)^mu

Dirac’s equation in quantionic formalism reads:

D |q) = i m Gamma^1 |q^* )

With D a right derivation quantion.

The Klein-Gordon equation in quantionic formalism reads:

(D’Alembert + m^2) |q) = 0

with |q) the column quantion.

Dirac went from Klein Gordon’s equation to a linear equation by talking the square root of the d’Alembertian:

(gamma^mu partial_mu) * (gamma^mu partial_mu) =   D’Alembert

quantions offer another decomposition of D’Alembert’s operator:

D^sharp D = D’Alembert

With D a derivarion right quantion and the sharp operation the parity-reversed transformation P of a quantion (quantions have the discrete CPT = I symmetry).

Quantionic time evolution is best understood in the gauge theory sense, but to wrap the standard quantum mechanics description, recall this from Hardy’s paper:

|mn> = 1/sqrt(2) (|m> + |n>)

|MN> = 1/sqrt(2) (|m> + i|n>)

and that there were

½ N(N-1) projectors of the form |mn><mn|

Because quantionic quantum mechanics has a non-commutative number system the corresponding most general projectors here is of the form:

|mn>P<mn|

|MN>P<MN|

with P^2=P and P a unit quantion. We know p+ = (1+sigma)/2 and  p- = (1-sigma)/2, are spin projector operators where sigma are the Pauli matrices.

From here it is not hard to show that are 4 linear independent unit quantionic projectors P (one for identity and 3 for each of the Pauli matrices) corresponding to the spin degree of freedom, or to the 4 spinor components. Hardy’s formula then reads:

K = 4N^2 for N quantions

An instructive exercise for the reader is to investigate why quaternionic quantum mechanics does not have projectors of this type (I’ll give the answer in the next post).

Next time I’ll finish the quantionic quantum mechanics presentations from the gauge theory point of view.

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What is the number system of quantum mechanics?                   

Quantionic quantum mechanics

So far we have seen that quantum mechanics can be expressed over real numbers, complex numbers, and quaternions. Physically, quantum mechanics over reals and quaternions do not lead to new predictions. Also this short list implies that quantum mechanics is about Born rule and information. We shall see that this is not the complete story.

Quantionic quantum mechanics is a direct counter example to arbitrarily restricting the number system to division algebras. Quantionic quantum mechanics was discovered by Emile Grgin (or Gergin) – I am proud to state that my “Grgin number” is one. Working in Peter Bergmann’s group, Grgin joined forces with Aage Petersen, Bohr’s personal assistant and investigated a line of thought from Bohr about the correspondence principle. This resulted in the composability principle and a dual quantum-classical mechanics framework (pre C* algebra) better than Segal’s quantum algebraic formalism. This work happened in early 70s and was forgotten after Grgin left academia and went to work for the industry. Recently Grgin retired and restarting working in this area resulting in quantionic quantum mechanics – a towering achievement which unfortunately is not well known.

Probably the best way to think quantionic quantum mechanics is terms of Darwin’s evolution: it is a “missing link” between regular quantum mechanics and gauge theory. When talking about quantions, one can either take the point of view of standard quantum mechanics, or the point of view of field theory. Its physical content is identical with Dirac’s theory of the electron, but its formalism is most illuminating.

So let start the story from the trusted Born’s rule. This implies that quantum mechanics is only about information. But is it? All experiments are done in space-time and events require 4 coordinate numbers (x,y,x,t) to be located. In turn Lorentz transformations and special relativity teaches us about 4-vectors, so in a relativistic quantum mechanics it is natural to think not of probabilities, but of probabilities currents. The starting point of quantionic quantum mechanics is a generalization of Born’s rule to probabilities currents (called the “Zovko interpretation”-after the person who discovered it).

Complex numbers have two norms, let’s call them A for algebraic and M for metric. In matrix representation a complex number z = a+ib correspond to a 2x2 matrix:

 a   b

-b  a

The algebraic norm A is defined as: A(z) = z^{dagger} z:

 a  -b     a  b   =  (a^2 + b^2)  1  0

 b   a    -b  a                           0  1

The metric norm is the determinant: M(z) = det (z) = a^2 + b^2

The value is the same, but the meaning is very different. Quantionic quantum mechanics aims to lift this degeneracy and have two fully distinct norms.

Quantions are based on the SL(2, C) ~ SO(3,1) isomorphism and are defined as follows:

q1  q3   0   0

q2  q4   0   0

0    0   q1  q3

0    0   q2  q4

with q1, q2, q3, q4 complex numbers. The multiplication rule for the algebra of quantions (q1, q2, q3, q4) follows from the matrix multiplication rule.

OK, this is a bit dry, and to put it in physics context, for quantions M correspond to relativity or with picking a particular frame of reference, and A corresponds to quntum mechanics and the inner product.

For a quantion Q:

A(Q) = Q^dagger Q :  a future oriented 4-vector

M(Q) = det(Q) : a complex number

The reason for the block diagonal zero elements have to do with gauge theory and quantion-spinor correspondence, and to simplify the idea, we can talk for now about a reduced quantion:

q1  q3

q2  q4

A hermitean reduced quation:

 r   z

z*  s

corresponds to a 4-vector: (p0, p1, p2, p3) in the Minkowski space as follows:

p0 + p3        p1+ i p2

p1 – i p2       p0 – p3

More important, A(Q) = Q^dagger Q is a future oriented 4-vector. (recall from prior posts that spin factors are realizations of the Jordan algebra)

The fundamental theorem of quantionic algebra is that the algebraic and metric norm commute: AM(Q) = MA(Q)

Pictorially this can be represented as:

        q   ----------------A------------------->A(q) = future oriented 4-vector

         |                                                       / |         on Minkowski space

         |                                                    /    |

p -------------------A------------------>A(p) = null vector

         |                                                  |       |                                                    

|        M                                               M     |

|        |                                                  |       M

M     |                                                  |        |

|       z   ---------------A----------------------->x  

|                                                           |    /    

|                                                           |  /

0 ------------------A---------------------> 0

with p a quantion of determinant zero

The 0-x axis represents the good old-fashion Born rule: the experimental predictions are positive probabilities.

A(q) inside the Minkowski cone A(p) represents Zovko interpretation as probability current.

But is quantionic quantum mechanics nothing but complex quantum mechanics with relativistic symmetry? No, the story is subtle (and this is where spin enters the picture).

In standard complex (non-relativistic) quantum mechanics, a state omega of a quantum mechanical system is a linear functional omega on the space of observables A with the property:

Omega (A^dagger A) >= 0 for all observables in A

Omega (I) = 1

The Hilbert space A has the inner product:

<A, B> = Tr (A^dagger B)

In quantionic quantum mechanics the inner product does not involve the trace. Here:

<A, B> = M (A^dagger B) = determinant (A^dagger B)

In turn this demands the existence of spin.

If I can coin a word, in complex quantum mechanics one talks of qubits, while in quantionic quantum mechanics one talks of z-bits (z from Zovko).

In quantionic quantum mechanics superposition is based on quantions. Recall Adler’s requirements for a modulus function N:

(1)        N(0) = 0

(2)        N(phi) > 0 if phi is not 0

(3)        N(r phi) = |r| N(phi)

(4)        N(phi_1 + phi_2) <= N(phi_1) + N(phi_2)

where N = AM in this case.

Now (2) has to be relaxed: there are quantions phi for which N(phi) = 0 and also (4) is not true in general.

The fact that (4) does not hold is not a problem because it does not hold in general for Hilbert modules either which generalize C* algebras for vector bundles (field theory). Quantionic quantum mechanics can be understood as electroweak gauge theory!!!

Naively, the relaxation of condition (2) can be treated with the tools of C* algebra in the usual GNS construction:

Here singular means that the sesquilinear form may fail to satisfy the non-degeneracy property of inner product. By the Cauchy–Schwarz inequality, the degenerate elements,x in A satisfying ρ(x* x)= 0, form a vector subspace I of A. By a C*-algebraic argument, one can show that I is a left ideal of A. The quotient space of the A by the vector subspace I is an inner product space. The Cauchy completion of A/I in the quotient norm is a Hilbert space H.

In quantionic quantum mechanics this quotient space is empty! This is another clue that there is something fundamentally different at play here and that the Jordan spin factor algebras, although embeddable in the nxn matrix case is a beast of its own.

From the zero determinant quantions, it is clear that quantionic algebra is a non-division associative algebra!!!

And why do we need division in quantum mechanics? It is silly to arbitrarily insist on a mathematical property just for our convenience. In what quantum mechanics book or paper have you seen any direct division by wavefunctions?

The beauty of quantionic quantum mechanics is that nothing needs to be postulated except Zovko’s interpretation. In subsequent posts we’ll see how Dirac equation follows naturally, we’ll investigate the spinor-quantionic correspondence, we’ll give a counter example for Hardy’s formula: K = N^r (for quantionic quantum mechanics the formula is K = 4 N^r), and we’ll present quantionic quantum mechanics from the gauge theory point of view.

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