Friday, January 23, 2015

Quantum Groups and curved space-time


Today I will present the last post on Hopf algebras and I will talk about quantum groups (which are a special kind of Hopf algebras) and their amazing application to curves space-time. Alfredo Iorio pointed me his paper (http://arxiv.org/pdf/hep-th/0104162v1.pdf) which I studied in detail.

In second quantization he starts with the Weyl-Heisenberg algebra \((a, a^{\dagger}, N, c)\):

\([a, a^{\dagger}] = 2c\)
\([N, a] = −a \)
\([N, a^{\dagger}] = a^{\dagger} \)
\([c, \cdot] = 0\)

and then the Hopf algebra coproduct is introduced:

\(\Delta a = a\otimes I + I \otimes a\)

So far nothing special but then the coproduct can be deformed forming a quantum group:

\(\Delta a_q = a_q \otimes q^c + q^{-c}\otimes a_q\)

where \(q\) is the deformation parameter related to a geometric-like series:

\({[x]}_q = \frac{q^x - q^{-x}}{q - q^-1}\)

As a nice math puzzle, what is the series which gives rise to \({[n]}_q\)? It is obviously related to \( 1+q+q^2+\cdots + q^n\) but it is not quite that.

So what is this to do with anything? 

The q-deformation o reproduces the typical structures of a quantized field in a space–time with horizon!!! 



Quantum group deformations can be induced by gravitational fields. For technical details please read Iorio's paper from above.

I am still studying quantum groups and I do not want to venture stating more because my intuition in this area is work in progress, but clearly they have a real physical interpretation.

Friday, January 16, 2015

My first physics paper


I was looking through a pile of old papers and I discovered a copy of my very first physics paper which I published in my 4th year in college - quite a long time ago. I remember I was attending a standard electromagnetism class and the teacher said: to obtain the invariants of the electromagnetic filed, compute the characteristic polynomial:

\( det(F -I_4 x) = 0\)

where \( F\) is electromagnetic field tensor. The coefficients are the invariants.

I went home and double checked the math and surprise: the equation implied:

\( x^4 -  (B^2 +E^2 ) x^2 + {(E\cdot B)}^2 = 0 \) 

So the next lecture I confronted the professor and showed that you get \(B^2 + E^2\) instead of \(B^2 - E^2\) and his statement was wrong. To my surprise he said that he knew it was wrong, but it was close to the answer and it must have a kernel of truth but he did not know how to fix it. Now in college I never liked functional analysis (and only recently I developed the right intuition in the area) but I was always good at algebra and I could come up with the answer to any algebra problem at first sight. So I said to the teacher: I know how to fix it: just make the electric field imaginary. The professor than said: OK, write it up and if you can do it we'll write a paper. 

This was not that easy -it took me an afternoon- but I worked it at home starting backwards. Suppose you have a \(4 \times 4\) diagonal matrix \(J\) with the diagonal elements \((1,1,1,i)\) Then if you multiply the electromagnetic tensor \(F\) with \(J\) on the left and on the right: \(J F J\) then you are in business with \( det(J F J -I_4 x) = 0\). But why does this work? What is happening is a transition from a pseudo-orthogonal group to an orthogonal group. Here is how:

The electromagnetic tensor changes under a Lorentz boost \( \Lambda \) like this:

\( F^{'} = \Lambda F {\Lambda}^t\)

where 

\({\Lambda}^t Y \Lambda = Y\)

with \(Y \) the diagonal matrix \((1,1,1,-1)\)

Now if we sandwich \( F \) with \( J\) we get:

\(J F^{'} J = J \Lambda F {\Lambda}^t J = (J \Lambda J^{-1}) (J F J) {(J \Lambda J^{-1})}^t \)

But what about \(\Lambda\) ? Here is the fireworks: \(Y = J J\) 

So let us compute \({(J \Lambda J^{-1})}^t (J \Lambda J^{-1}) \):

\({(J \Lambda J^{-1})}^t J \Lambda J^{-1} = J^{-1} {\Lambda}^t J J \Lambda J^{-1} = J^{-1} {\Lambda}^t Y \Lambda J^{-1} = \)

\( = J^{-1} Y J^{-1} = J^{-1} J J J^{-1} = 1\)

Bingo: we manage to get an orthogonal transformation from a pseudo-orthogonal one. This is reminiscent of Dirac's trick because we basically take the square root of \(Y\) and we get two \(J \) matrix instead. This is how people used to write imaginary \(ict \) time and preserve the usual orthogonal rotations. And for orthogonal group there is an elementary linear algebra theorem which states that the only invariants of a similarity transformation (which is how the electromagnetic tensor changes under a Lorentz transformation in the new complexified orthogonal transformation) are the coefficients of the characteristic polynomial. 

So lo and behold, my first physics paper appeared in the Romanian Journal of Physics, Volume 38, Number 9, pages 873-875 in 1993. The teacher's name was Andrei Ludu and he was also doing seminars on quantum groups. At that time I had no clue on Lie algebras, let alone on quantum groups and I could not find any motivation or intuition to a bunch of very long and way too abstract things. 

But guess what? Quantum groups are actually Hopf algebras and they have very interesting physics applications. I'll talk about it next time.

Saturday, January 10, 2015

Renormalization and Hopf algebras


In quantum electrodynamics perturbation analysis leads to infinities for the mass and the charge of the electron due to self-energy Feynman loop diagrams. Those infinities can be isolated and cured by a renormalization process. However renormalization is much more widespread than one may think. Sidney Coleman suggested a simple experiment one can do at home to see it in action. Take a ping-pong ball and immerse it water. Compute the Archimedean force and see that it should produce an acceleration close to an amazing 11 g. At that acceleration the ping pong ball should shoot out of the water and hit the ceiling, but of course this does not happen. What happens is that the acceleration is in fact close to 2 g because the mass of the ball that should be used is an "effective mass" due to the surrounding fluid. 

Now in quantum field theory there are many renormalization techniques possible. Connes and Kreimer have had a truly outstanding and amazing result relating the combinatorial Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization technique with Hopf algebras and the Birkhoff factorization problem. I cannot do justice to this topic in a simple post, and I encourage the interested reader to study Connes' book: "Noncommutative Geometry, Quantum Fields and Motives" for the details. I will only present the big picture here.

The first concept needed is what Connes calls: one-particle irreducible Feynman graphs (1PI) \( \Gamma\):
  • \( \Gamma \) is not a tree
  • \( \Gamma \) cannot be disconnected by cutting a single edge
Then given a graph \( \Gamma \) it may have a 1PI subgraph \( \gamma \) and one can construct a contracted graph called \( \Gamma / \gamma\) which is obtained by collapsing each \( \gamma_i \) - 1PI of \( \Gamma\) to a vertex. 

Now the coproduct of the Hopf algebra of the Feynman graphs is given by:

\( \Delta (\Gamma) = \Gamma \otimes 1 + 1\otimes \Gamma + \sum_{\gamma} \gamma \otimes \Gamma / \gamma \)

Image from http://arxiv.org/abs/hep-th/9912092

The coalgebra so obtained is graded by the loop numbers, and so it is in fact a Hopf algebra. 

Now onto the Birkhoff decomposition. Start with the Riemann sphere and consider a closed loop \( \gamma \) on it. The loop cuts the sphere into two separate regions + and -. You are given a complex value function \( \phi \) along the loop and now we want to find two complex functions \( \phi_{+} \) defined on the + region and \( \phi_{-} \) defined on the - region such that \( \phi = {\phi_-}^{-1}  {\phi_+}\). In passing we note that this decomposition problem is intimately related with soliton theory and inverse scattering

Now for the main result. First the BPHZ procedure: given a Feynman diagram \( \Gamma \) one performs the Bogoliubov-Parasiuk preparation: replace the unrenormalized value \(U(\Gamma)\) by:

\( R(\Gamma) = U(\Gamma ) + \sum_{\gamma} C(\gamma) U(\Gamma / \gamma)\)
where the counterterms \(C(\gamma)\) are defined recursively as:

\(C(\Gamma) = -T(R(\Gamma))\) where \(T\) is the projection onto the pole part of the Laurent series in the variable z of dimensional regularization.

Now if we have a renormalizable theory and its associated Hopf algebra of Feynman diagrams defined above the Birkhoff factorization is given by:

\( \phi_- (X) = -T (\phi (X) + \sum\phi_- ( X^{'}) \phi ({X}^{''})) \) 
and
\(\phi_+ (X) = \phi (X) + \phi_- (X) + \sum \phi_- ({X}^{'}) \phi ({X}^{''} )\)
where the coproduct is:
\( \Delta (X) = X\otimes 1 + 1\otimes X + \sum {X}^{'} \otimes {X}^{''}\)

I hope now thing become clear: we start with dimensional regularization which extends the dimension of space into complex numbers (in fact into the Riemann sphere). The unrenormalized values corresponds to a function \(\phi\) on a closed loop in the Riemann sphere and we use the Hopf algebra of Feynman diagrams to perform the Birkhoff factorization which obtains the renormalized values \(\phi_+ \).

BPHZ is the same as Birkhoff factorization:

\(\phi = U\) - unrenormalized value
\(\phi_- = C \) -counterterms
\(\phi_+ = R \) - renormalized value

The Hopf algebra coproduct simply codifies the BPHZ procedure. Whow!!!! What a mathematical tour-de-force linking such unrelated mathematical areas. I don't know about you but I am really impressed with Connes and Kreimer result. Bravo!

Saturday, January 3, 2015

The Composability Interpretation


Before discussing the Hopf algebra application to renormalization, I want to share the draft of a document I am preparing on a new quantum mechanics interpretation. Please feel free to criticize it. Enjoy!

Abstract

Quantum mechanics is described by a set of mathematical structures: Hilbert space, the commutator, the symmetrized product of Hermitian operators. When two quantum systems are combined the mathematical formalism remains the same. For example the tensor product of two Hilbert spaces is still a Hilbert space and observables are still described by self-adjoint operators. Two quantum mechanics systems cannot be combined to generate a classical mechanics system. Very few mathematical structures can obey this self-similarity invariance under tensor composition. It turns out that invariance under tensor composition along with additional natural assumptions completely determine all algebraic properties of quantum mechanics. In turn this introduces a new quantum mechanics interpretation: the composability interpretation.

1. Quantum Mechanics Reconstruction


Quantum mechanics is defined by a series of technical axioms which lack a clear physical justification. For example, why should the state of a quantum system be described as a vector in a Hilbert space? Or why should observables be represented by Hermitean operators? Also quantum mechanics exhibits a series of counterintuitive behaviors which are consequences of the mathematical structure of the theory. One such behavior is the existence of quantum correlations which exceeds the maximum possible classical correlations. The maximum possible quantum correlation, known as the Tsirelson bound, is a consequence of operator's norm in a Hilbert space [1].

The special theory of relativity also exhibits unintuitive phenomena like the relativity of the simultaneity, length contraction, or time dilation. However, in this case we do not attempt to understand directly those effects but instead we understand the principles of the special theory of relativity. A natural question to ask is then: can quantum mechanics be obtained as a principle theory?  If such an approach is possible then the interpretation of quantum mechanics follows naturally from the physical axioms.

The first hurdle to a reconstruction approach is the multitude of quantum mechanics formulations. Nonrelativistic (no spin) quantum mechanics can be formulated the usual way in the Hilbert space formalism or in the phase space formalism. Yet another approach is the path integral formalism. In the late 70's Emile Grgin and Aage Petersen studied the common mathematical structure of quantum and classical mechanics and introduced a so-called two-product algebra [2] which now is called as a Jordan-Lie algebra [3]. Sometimes the definition of a Jordan-Lie algebra contains also norm properties [4]. This is an algebraic formulation of quantum mechanics which in particular cases can lead to the usual C*-algebra formulation of quantum mechanics by deformation quantization. The advantage of this formulation is twofold: first the mathematical structure of two-product algebra is distinguished by virtue of being common to both quantum and classical mechanics, and second the mathematical properties of the two-product algebra can naturally be obtained from physical principles. The full quantum mechanics reconstruction from the two-product algebra formalism is a work in progress but the results available so far will allow us to introduce a new quantum mechanics interpretation. We will start by presenting the two-product algebra mathematical structures and their physical interpretation. For reasons which will later become apparent we will call the two-product algebra "the composition algebra".

2. The Composition Algebra


Up to norm properties, the composition algebra is the algebraic structure of quantum mechanics. There are three kinds of composition algebras which we will call elliptic, parabolic, and hyperbolic. Quantum mechanics corresponds to the elliptic case, while classical mechanics corresponds to the parabolic case. The hyperbolic case corresponds to the so-called hyperbolic quantum mechanics [5].

For the finite dimensional case the elliptic composition algebra completely recovers the Hilbert space structure [6]. For the general case we need to augment it with positivity as a way to guarantee the rejection of the hyperbolic case and the existence of deformation quantization. Reconstruction of the quantum cases which have no classical correspondence is a work in progress.

2.1. Components of the Composition Algebra


The (elliptic) composition algebra consists of the Jordan algebra of observables, the Lie algebra of generators, and their compatibility relationship. Loosely speaking, observables correspond to ontology and generators to dynamics. Quantum mechanics has a nontrivial compatibility relationship which is the root cause of its peculiar behavior.

States do not enter discussion at this point. An older axiomatization attempt by Segal from 1947 [7] considers the duality between observables and states but this goes beyond physics because it allows the algebra not to be involutive. In turn this allows the dynamic to be generalized into something we do not see in nature because the Lie algebra product is replaced by a product which is not skew-symmetric.

In the composition approach there is a duality between observables and generators and this is related with Noether theorem: observables are generators of kinematics symmetries and an observable which is preserved by time evolution generates a dynamical continuous symmetry [6].

States arise out of the compatibility condition which enables the definition of an associative product. Associativity is a required property for the algebra of states.

2.1.1 The Jordan Algebra

In quantum mechanics observables are Hermitean operators \(O^\dagger = O\), but the product of two operators does not preserve the hermiticity condition. The Jordan product \(\sigma\) is defined as the symmetrized product: \(A\sigma B = \frac{1}{2} (AB+BA)\) and this preserves hermiticity meaning that \(A\sigma B\) is another observable. In general the Jordan product is not associative but power associative and satisfies the Jordan identity: \((A\sigma B) \sigma (A\sigma A) = A\sigma (B\sigma (A \sigma A))\).

Quantum mechanics admits two points of view: the usual one based on states, and an instrumentalist one based on observables. The Jordan algebra can be either the self-adjoined part of a C*-algebra or can be special. There is only one special algebra called the Albert algebra and this is the set of \(3\times 3\) self-adjoint matrices over the octonions. Albert algebra does not play a physical role because the compatibility relationship between the Jordan and Lie algebras rejects special Jordan algebras. However, all other Jordan algebras over reals, complex numbers, quaternions, or spin factors can be used to define quantum mechanics over different number systems.

2.1.2 The Lie Algebra

In mathematics Lie groups are continuous groups and they typically describe symmetries. Because of continuity, in the neighborhood of the identity one can define a tangent plane and this gives rise to a Lie algebra. Elements of a Lie algebra are called "generators" and they can be thought of as the infinitesimal vectors generating the group by using the exponential map. A Lie algebra is given by a skew-symmetric product obeying the Jacobi identity.

In the Heisenberg picture the time evolution is described using the commutator which can be understood as a skew-symmetric product \(\alpha\): \(A \alpha B = \frac{J}{\hbar} [A,B]\) where \(J^2 = -1\). Moreover the product \(\alpha\) obeys the Jacobi identity: \(A\alpha (B\alpha C) + C\alpha (A\alpha B) + B\alpha (C\alpha A) = 0\) and as such it forms a Lie algebra.

Because time evolution comes from the commutator, \([H, A]\) is the Lie derivative of A along the Hamiltonian vector field generated by \(H\). Therefore in general the product \(\alpha\) is a derivative obeying the Leibniz identity: \(A\alpha(B\circ C) = (A\alpha B) \circ C + B\circ (A\alpha B)\) for any product \(\circ\). In particular when \(\circ = \alpha\), due to skew-symmetry, the Leibniz identity reduces itself to the Jacobi identity.

2.1.3 The Compatibility Condition

In quantum mechanics there is a one-to-one correspondence between observables and generators because observables are hermitean operators and generators are anti-hermitean operators. As such there is a linear map \(J\) between the two spaces. Also there is a compatibility condition between the two products \(\alpha\) and \(\sigma\) as follows: \((A\sigma (B \sigma C) - (A \sigma B) \sigma C) + \frac{J^2 \hbar^2}{4} (A\alpha (B \alpha C) - (A \alpha B) \alpha C) = 0\). Then one can combine the two products \(\alpha\) and \(\sigma\) into an associative product which is nothing but the usual complex number multiplication for operators on the Hilbert space: \(* = \sigma + \frac{J \hbar}{2}\alpha\).

For any product \(\circ\), \((A\circ (B \circ C) - (A \circ B) \circ C)\) is called the "associator": \([A, B, C]_{\circ}\) and it quantifies the violation of associativity. The compatibility condition is then: \([A, B, C]_{\sigma} + \frac{J^2 \hbar^2}{4} [A, B, C]_{\alpha} = 0\) and it is this relationship which excludes the existence of octonionic quantum mechanics.

2.1.4 Comparison with Classical Mechanics

It is very enlightening to note that classical mechanics has the same composition algebra structure where the product \(\alpha\) is the Poisson bracket and the product \(\sigma\) is the regular function multiplication. The only difference is in the map \(J\) which for classical mechanics obeys \(J^2 = 0\). For classical mechanics the composition algebra is known as the Poisson algebra.

Because \(J^2 = 0\) we have from the compatibility condition that \([A, B, C]_{\sigma} = 0\). This allows a state space to be defined directly for the observables which have "objective reality". As a consequence in classical mechanics non-pure states have an ignorance interpretation. However the ignorance interpretation is untenable for quantum mechanics.

2.2. Composing Two Systems


If quantum mechanics' algebraic structure is given in terms a symmetric product (the Jordan product) and a skew-symmetric product (the commutator) we can ask what happens when we compose two quantum systems into a larger one. The larger system also has a symmetric and a skew symmetric product and this must be built from the original ingredients. This is an essential point because if other mathematical structures would be allowed then quantum theory would not be a fundamental theory.

Suppose we call the products \(\sigma_1, \alpha_1\) for the first quantum system and \(\sigma_2, \alpha_2\) for the second quantum system. By symmetry property the only way we can combine the four products \(\sigma_1, \alpha_1, \sigma_2, \alpha_2\) to generate a symmetric and a skew-symmetric product is as follows:

\begin{eqnarray}
\sigma_{1\otimes 2} = a \sigma_1 \otimes \sigma_2 + b \alpha_1 \otimes \alpha_2 \\
\alpha_{1\otimes 2} = c \alpha_1 \otimes \sigma_2 + d \sigma_1 \otimes \alpha_2 \nonumber
\end{eqnarray}

The parameters \(a, b, c, d\) have the following values: \(a=c=d=1\) and \(b = \frac{J^2 \hbar^2}{4}\).

In the quantum mechanics case \(b = -\frac{\hbar^2}{4}\) and we see that one cannot have a factorization of the Jordan product using only Jordan products. In turn this shows that quantum mechanics does not obey the so-called "Bell locality" condition [8] and measurements on a system have an effect on a remote system because observables on a composite system depends on more than the subsystems' observables.  

For classical mechanics \(b = 0\) while for hyperbolic quantum mechanics \(b = +\frac{\hbar^2}{4}\).

Please note the similarity of the composition formula in the quantum case with the complex number multiplication. This similarity explains why quantum mechanics is most naturally described using complex numbers. Similarly hyperbolic quantum mechanics is best described in a split-complex numbers formalism.

From the composition algebra one can easily extract the phase space representation. For quantum mechanics this comes in the form of cosine and sine Moyal brackets [9]:

\begin{eqnarray}
\alpha &=& \frac{2}{\hbar}\sin ( \frac{\hbar}{2}\overleftrightarrow{\nabla}) ,\\ \nonumber
\sigma &=& \cos (\frac{\hbar}{2}\overleftrightarrow{\nabla}) ,
\end{eqnarray}

where the operator \(\overleftrightarrow{\nabla}\) is defined as follows:

\begin{equation}
\overleftrightarrow{\nabla} = \sum_{i=1}^{N} [ \overleftarrow{\frac{\partial}{\partial x_i}}\overrightarrow{\frac{\partial}{\partial p_i}} - \overleftarrow{\frac{\partial}{\partial p_i}}\overrightarrow{\frac{\partial}{\partial x_i}}] .
\end{equation}

Then one can easily verify that Eqs. 2 are a realization of the composition algebra. From the phase space representation one arrives at the usual Hilbert space representation by Berezin quantization [10].

The Berezin quantization is the following prescription to construct compact operators from continuous functions on phase space:

\begin{equation}
Q_{\hbar} (f) = \int_{{\bf R}^{2n}} \frac{dp dq}{2 \pi \hbar} f(p,q) |\Phi_{\hbar}^{(p,q)} \rangle \langle \Phi_{\hbar}^{(p,q)} | ,
\end{equation}

where \(\Phi_{\hbar}^{(p,q)}\) are coherent states defined as:

\begin{equation}
\Phi_{\hbar}^{(p,q)} = {(\pi \hbar)}^{-1/4} e^{-ipq/2 \hbar} e^{ipx/\hbar} e^{-{(x-q)}^2 /2 \hbar} .
\end{equation}

Essential in the Berezin quantization is positivity and this has an additional benefit: hyperbolic quantum mechanics does not respect positivity and hence is rejected by it. Hyperbolic quantum mechanics is unphysical because one cannot construct a state space able to generate non-negative probability predictions and one cannot attach any interpretation to it.

As a side note we point out that both classical and quantum mechanics can be expressed in either phase space or a Hilbert space formalism [11] but for classical mechanics we stay in the usual phase space formalism.

There are two more points to discuss. First, even though there is a Hilbert space formulation for classical mechanics, there is no superposition in this case. Also, superposition is not "the characteristics of quantum mechanics" because superposition is present in hyperbolic quantum mechanics as well [5]. The quantum wavefunction \(\Psi\) is just a tool for computing probabilities according to Born rule. Second, the number system for quantum mechanics can generalize Born's rule. The number system originates from the classification of real Jordan algebras and takes the form of either reals, complex numbers, quaternions, or spin factors. In the first three cases quantum mechanics predictions are probabilities related to the norm of the division number systems. In the spin factor case, a less known realization of quantum mechanics over \(SL(2,C)\) is possible and this is equivalent with Dirac's theory of the electron. Here the predictions are not probabilities but Dirac's 4-current probability density and the C*-algebraic formulation generalizes to a C*-Hilbert module. All those are advanced topics and in the following we will confine ourselves only to the standard quantum mechanics over complex numbers.

We have seen that the quantum formalism follows from the composition algebra and now we can ask the critical question: can we derive the composition algebra form physical principles? The answer is yes and involves the composition relationships of Eqs. 1 in an essential way.

2.3. Reconstructing the Composition Algebra


The mathematical formalism required is that of category theory. Informally, category theory is known as "objects with arrows" and the idea is to change the point of view from sets and their elements to objects and morphisms. The set theory paradigm fits best with classical intuition and Bell locality factorization but the categorical approach is better suited to quantum mechanics.

Grgin and Petersen introduced the idea of composing two physical systems into a larger one and mathematically this gives rise to a commutative monoid. A commutative monoid respects three properties: commutativity, associativity, and the existence of a unit. Physically, associativity and commutativity means that that it does not matter how we compose the subsystems into a larger one. In turns this demands that the Planck constant is indeed a constant. The existence of the unit comes from the fact that the laws of nature are relational and the ground energy level does not affect the dynamics.

The composition algebra can be constructively reconstructed from the following physical principles:

  1. Invariance of the laws of nature under time evolution
  2. Invariance of the laws of nature under tensor composition
  3. The laws of nature are relational

The invariance of the laws of nature under time evolution means that given a time translation operator \(T\) one has:

\begin{equation}
T(A\circ B) = T(A) \circ T(B)
\end{equation}

where \(\circ\) can be any algebraic product (for example Poisson bracket, Jordan product, commutator, etc). Infinitesimal time translations leads to the product \(\alpha\). Because of the relationality of the laws of nature, a single product \(\alpha\) is not enough and a second product \(\sigma\) is required.

Together it can be shown that they obey the composition relationship of Eqs. 1 but without any symmetry properties yet. Again using relationality one can determine the products' symmetry properties and their compatibility relationship [2]. There is only one free parameter in the form of \(J^2\) which can be minus one for quantum mechanics (elliptic composition), zero for classical mechanics (parabolic composition) or plus one for hyperbolic quantum mechanics (hyperbolic composition). The overall factor of \(\frac{\hbar^2}{4}\) is introduced just to normalize the products to their usual definition as commutator and Jordan product. In the hyperbolic case one cannot construct a state space able to generate positive probability predictions and hence this case is unphysical. The distinction between classical and quantum mechanics stems from the violation of Bell's inequalities. Nature can only be in one of the three composition classes.

Invariance of the laws of nature under tensor composition means that the dynamical laws do not change their form when additional degrees of freedom are considered.

From the composition algebra, the Hilbert space formulation can be obtained by deformation quantization (for example by Berezin quantization [10] but other quantization methods are possible) and this enables the extraction of the norm properties for bounded operators which completely recovers the C*-algebraic formulation.

3. The Composability Interpretation


3.1 Is the Wavefunction Ontological?


If the wavefunction is ontological we would expect uniqueness of the mathematical description. However there are several distinct mathematical realizations of the composition algebra.

For example the domain of the two products \(\alpha\) and \(\sigma\) can be either operators in a Hilbert space, or functions in a phase space. In phase space the functions are continuous, but in a Hilbert space the functions are analytical.

Even in the Hilbert space representation, there is freedom to choose the number system for quantum mechanics. Quaternionic quantum mechanics makes the same predictions as complex quantum mechanics [12], but the quaternionic wavefunction is distinct from the complex wavefunction and the inner product is distinct as well. Moreover, for the infinite degrees of freedom case of quantum field theory a Hilbert space may not even exist although this is usually cured by introducing periodic boundary conditions to eliminate the infrared problem.

From an instrumentalist point of view we can understand a state as an equivalence class of preparation procedure, but its actual mathematical representation lacks uniqueness. Because all representations produce the same experimental predictions there is no experimental test possible to distinguish one representation from another.

All this points to a single conclusion: the usual wavefunction \(\Psi\) cannot be ontic.

3.2. Is the Wavefunction Epistemic?


In quantum mechanics there are two kinds of time evolution. First there is the unitary evolution which stems from the product \(\alpha\). Then there is a discontinuous time evolution after measurement. The reduction of the quantum wavefunction after measurement has a very simple explanation in terms of information update in an epistemic interpretation.

However, there is a major mathematical problem with discontinuous time evolution. Because the hermitean operators and the Hilbert space can be derived from unitary time evolution, violation of unitarity renders the entire mathematical formalism mathematically inconsistent. So can the wavefunction reduction be derived using only continuous time evolution? At first this seems absurd but it can be done in the categorical formalism.

We start by considering the commutative composition monoid introduced by Grgin and Petersen. Wavefunction reduction corresponds to an inverse operation which upgrades the monoid to a group. In category theory there is an universal property called the Grothendieck construction which builds a group from a monoid using an equivalence relationship [13]. This is best illustrated by showing how to construct the group of integer numbers starting from the monoid of positive numbers. Usually we say that there is a mathematical operation called subtraction and the negative numbers are what remain after subtracting a higher positive number from a smaller positive number. In what follows please keep in mind the similarity: "subtraction - collapse". We want to construct the negative numbers without introducing a new operation and we want to build the projection postulate in quantum mechanics while avoiding the non-unitary collapse evolution.

Here is how one proceeds in the case of integer numbers: construct a Cartesian product of two positive numbers and call the first element positive and the second element negative: \((A_p, A_n)\). Then observe that two Cartesian products \((A_p, A_n)\) and \((B_p, B_n)\) are equivalent if \(A_p+B_n + k = A_n + B_p + k\). For example the element \(-3\) is the following set of equivalent Cartesian products: \(-3 = (0, 3) = (1, 4) = \cdots\)

Because this construction is universal it can be applied to any monoid provided that we have a similar equivalence relationship. In elliptic composability there is a natural equivalence relationship which comes from envariance: what the system can unitarily evolve over here can be undone by another unitary evolution of the environment over there [14]. Parabolic composability (classical mechanics) has no such equivalence relationship.

We call two pairs of a Cartesian product of wavefunctions equivalent:

\begin{equation}
( {\left| \psi \right>}_p , {\left| \psi \right>}_n) \sim ({\left| \phi \right>}_p , {\left| \phi \right>}_n)
\end{equation}

if given any unitary transformation \(U_p\) acting on the left element \((\left| \psi \right>, \cdot )\) there exists a unitary transformation \(U_n\) acting on the right element \((\cdot, \left| \psi \right> )\), a wavefunction \(\left| \xi \right>\), and a unitary transformation \(U_{\xi}\) such that:

\begin{equation}
{\left| \psi \right>}_p \otimes {\left| \phi \right>}_n \otimes \left| \xi \right> =
(U_{p} {\left| \phi \right>}_p ) \otimes (U_n {\left| \psi \right>}_n) \otimes (U_{\xi} \left| \xi \right>) .
\end{equation}

Then it is relatively straightforward to show that this relationship respects the usual properties of reflexivity, symmetry, and transitivity. The "positive" part of the Cartesian product represents the quantum system, while the "negative" part of the Cartesian product represents the apparatus and the environment. During measurement and the "wavefunction collapse" the information is not lost, but it is simply transferred to the environment's degrees of freedom.

3.2.1. The Measurement Problem

To fully solve the measurement problem we need to prove the existence of a unique outcome. Mathematically this corresponds to breaking the envariance's equivalence relationship. Although this can be done in a few cases in a mathematically exact way, at this time we lack a universal mechanism. Because the lesson of quantum mechanics reconstruction is that there is only pure unitary time evolution we need to consider the many-worlds interpretation solution to the measurement problem. However it is well known that in this case there is a basis ambiguity problem which requires the help of decoherence. The trouble with this approach is that the world split happens only after decoherence takes place and this is a completely unjustified assumption. It is much more natural to consider instead spontaneous collapse models. However they represent a testable departure from quantum mechanics and they are contrary to the spirit of the reconstruction approaches. Future experiments will be able to confirm or reject spontaneous collapse models.

No known solution to the measurement problem is acceptable. The hope is that the category theory formalism will help prove the universality of the breaking the envariance's equivalence relationship. The pure unitary time evolution implies that the wavefunction \(\Psi\) is not epistemic either.

3.3 The Composability-Dependent Concepts


What the two sections above showed is that quantum mechanics is neither ontic nor epistemic in the usual sense and this suggests that the ontic and epistemic concepts are dependent on the composition class. Can we find other examples of concepts which are not clear cut when applied to quantum mechanics?

For example consider the question of determinism. Is quantum mechanics deterministic? If we understand determinism to mean a unique outcome, the answer is obviously no due to operator noncommutativity. However quantum mechanics time evolution is pure unitary and this is fully deterministic, but this does not translates into predictable outcomes.

Does counterfactual definiteness hold in quantum mechanics? For the outcome results it does not, but for the operator algebras which are the core structures of quantum mechanics counterfactual definiteness does hold.

Seems that realism, epistemic interpretation, determinism, and counterfactual definiteness are a parabolic composability concepts justified by Bell's locality factorization and are not well suited for quantum mechanics.

3.4. Other Quantum Mechanics Properties


How about other common interpretation questions related to quantum mechanics?
Are there hidden variables? The answer is no because it is mathematically inconsistent to have mixed composability classes. A mixed quantum-classical system would lack the back reaction from the quantum onto the classical system [15].

Does the wavefunction collapses? This is an apparent (or to quote Bell: "for all practical purposes") effect and an abuse of mathematical language stemming from ignoring an additional mathematical structure: the Grothendieck group of composition. This is similar with special theory of relativity when one talks about "imaginary \(ict\) time" because of ignoring the metric tensor. Time is not an imaginary distance and the wavefunction does not collapse.

Is quantum mechanics local or nonlocal? Bell's analysis of the EPR incompleteness claim showed that quantum mechanics violates the so-called "Bell locality" and this is usually interpreted as saying the quantum mechanics is non-local. However, Bell's locality is coming from the parabolic factorization of the symmetric product \(\sigma\) and this has nothing to do with space-time. A better explanation is to state that quantum mechanics is locality-independent, or locality neutral.

4. Principles of quantum mechanics


Let us discuss the physical principles used to reconstruct quantum mechanics. Invariance of the laws of nature under time evolution is self-evident. We assume the existence of time when deriving quantum mechanics. However there is the so-called "thermal time hypothesis" [16] which is based on the Tomita-Takesaki theory which shows the existence of a distinguished one-dimensional parameter which can be understood as time [17].

Time seems to have a noncommutative origin and now we face a problem of circularity: time implies quantum mechanics and quantum mechanics implies time. One possible way out is to start not from the existence of time and invariance of the laws of nature under time evolution but to assume the existence of the Leibniz identity. Mathematically the Leibniz identity is valid in both commutative and non-commutative settings and this is a very helpful. However in this case we lose the physical intuition. As of now the apparent circularity is an open problem.

The key principle of quantum mechanics is that of invariance under tensor composition. This principle has several justifications. First, it means that the dynamics does not change with an increase in the numbers of degrees of freedom. Second, it implies that there are no "island universes" and the laws of nature remain the same regardless of how we partition a physical system in our minds. Third, it shows that quantum mechanics is the fundamental irreducible theory describing nature and that there are no sub-quantum explanations possible.

The reason composability principle is extremely powerful in reconstructing both classical and quantum mechanics is that it imposes a self-similarity condition on the mathematical structures and by category theory this completely determines the algebraic structure. In the finite dimensional case this is enough to completely reconstruct the theory.

Next, the relationality of the laws of nature generalizes the kinematics principle of relativity into the dynamic realm because the ground energy level does not affect the dynamics. Mathematically this corresponds to unitality in a bialgebra setting. A bialgebra is an algebra which has both a product and a coproduct. Technically the composition algebra is a two-product bialgebra with the products given by the Lie and Jordan products, and the coproducts given by the compositions of Eqs. 1.

Last, in addition to the three principles above we need positivity to eliminate the hyperbolic composability case. Positivity can be understood as the way to introduce an information requirement into quantum reconstruction program. Information is a characteristic of the real world which distinguishes it from the abstract world of mathematics. Physics is an experimental science and we should be able to extract information from the real world and make testable predictions. But information considerations are not enough to recover quantum mechanics and we can say that "it from bit" is not a valid strategy. Instead we should say: "it is what can generate a bit".

5. Conclusion


In the composability interpretation, the Hilbert space, the wavefunction, and the superposition principle are secondary derived concepts. The algebraic structure is fundamental and this follows from the physical principles used to reconstruct quantum mechanics. In a physical principle theory, the principles act differently than technical axioms: the principles are requirements which reject almost all mathematical structures with the exception of a distinguished few. In this case they select the composition algebra. Once the composition algebra is obtained, this represents an alternative mathematical formulation of quantum mechanics from which the standard formalism is recovered.

Quantum mechanics expands our understanding: from Boolean algebra to non-Boolean algebra, from commutative formalism to non-commutative formalism, from set theory to category theory. The category formalism includes set theory and allows us to specify mathematical relationships while avoiding questions about the nature of the ingredients. The usual concepts of realism or epistemology are ill suited to describe quantum mechanics.

The problem is not explaining quantum mechanics in terms of parabolic composability inspired concepts but explaining the emergence of the classical world from quantum mechanics.

References

[1] A. Peres, Quantum Theory: Concepts and Methods (Fundamental Theories of Physics), (Kluwer, Dordrecht, 1995), pp. 174-174
[2] E. Grgin, A. Petersen, Comm. Math. Phys. 50
[3] N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, (Springer, New York, 1998), pp. 37-41
[4] E. M. Alfsen, F. W. Shultz, Geometry of State Spaces of Operator Algebras (Birkhauser, Boston, 2003)
[5] A. Khrennikov, G. Segre, Int. J. Theor. Phys. 45
[6] A. Kapustin, J. Math. Phys. 54
[7] I. E. Segal, Ann. Math. 48
[8] J. S.Bell, Phys. 1
[9] J. E. Moyal, M. S. Bartlett, Math. Proc. Cambridge Phil. Soc. 45
[10] F. A. Berezin, Commun. Math. Phys. 40
[11] B. O. Koopman, Proc. Nat. Acad. Sci. 17
[12] S. L. Adler,  Quaternionic Quantum Mechanics and Quantum Fields (Oxford Univ. Press, New York, 1995)
[13] M. F. Atiyah, K-Theory, (Notes taken by D. W. Anderson, Fall 1964) ( W. A. Benjamin Inc., New York, 1967)
[14] W. H. Zurek, Phys. Rev. A. 71 
[15] D. Sahoo, J. Phys. A 37
[16] A. Connes, C. Rovelli, Class. Quant. Grav. 11
[17] A. Connes, Noncommutative Geometry (Acad.~Press, San Diego, 1994)

Wednesday, December 31, 2014

Happy New Year 2015!


The exploration of physics and math will continue in 2015. But now is a time to celebrate the new year. Happy New Year!


Thursday, December 25, 2014

The Hopf Algebra


Continuing the discussion, a bialgebra is a structure which is both an algebra and a coalgebra subject to a compatibility condition. A Hopf algebra H is a bialgebra with yet another property: the antipode. The antipode is a map from H to H and is usually named S.

If the bialgebra is a graded space, then the antipode comes for free and by an abuse of notation people call bialgebras Hopf algebras.

The antipode must be compatible with the existing structures of multiplication and comultiplication and so the diagram bellow commutes.



One thing I forgot to mention last time is that the unit "u" has a dual: the counit \( \epsilon\) which maps elements from the algebra to the field \( k \). Most of the time the counit maps everything to zero and occasionally to one.

Let us verify the commutativity of the diagram with our friend \(kG \). Here the antipode is the group inverse: \( S(g) = g^{-1}\):

Start from the left side: \( g\) 

Moving up and right: \( g \rightarrow g\otimes g\) then moving to the right and down: \( g \rightarrow g\otimes g \rightarrow\ g^{-1} \otimes g \rightarrow 1_H\) 

Now move from left to right on the middle line. If \( \epsilon \) maps all elements to \(1_K \) we have:

\( g \rightarrow 1_K \rightarrow 1_H \) and the diagram commutes. 

For a graded bialgebra the antipode is given by the following explicit formula:

\( S = \sum_{n \geq 0} {(-1)}^n m^{n-1} {\pi}^{\otimes n} {\Delta}^{n-1}\)

where \( \pi = I - u \epsilon \)

Next time we will see Hopf algebra application to renormalization in quantum field theory. If you want to read about Hopf algebras, the standard book is a 1969 book "Hopf Algebras" by Moss Sweedler who is known for the so-called Sweedler notation. Personally I do not like the style of the book because you get lost into irrelevant details and miss the forest because of the trees, but it is a good reference.

Wednesday, December 17, 2014

Coalgebras


Last time we introduced the coproduct which is the essential ingredient of a co-algebra. How can we understand it? If we think of the product as a machine which eats two numbers and generates another one we understand the coproduct as the same machine working in reverse. A Xerox machine can be understood as a coproduct, but a coproduct can be understood not only a a cloning machine but as an action which breaks up an elements into sub-elements. For example a complex number can be decomposed into a real and an imaginary part and each of those are nothing but other kinds of complex numbers.

One funny example comes from shuffling cards: cutting a deck of cards in two is the coproduct, while putting it back together in all possible ways is the product. Renormalization techniques in quantum field theory generates coproducts. Here is a partial list of well studied mathematical examples. The coproduct is usually expressed with the symbol \( \Delta \) and the product is represented by the symbol \(m\).

The first (and most trivial) example come from group theory. Consider finite linear combinations of group elements:

\( kG = \{ \sum_{i=1}^n \alpha_i g_i | \alpha_i \in k, g_i \in G\}\)

\(\Delta g = g\otimes g\)

This is nothing but a basic cloning operation. A bit more complex example comes from polynomial rings:

\( \Delta (x^n) = \sum_{i=0}^n (n~choose~i) x^i \otimes x^j \)
\(m(x^i \otimes x^j) = x^{i+j}\)

Much fancier example come from the cohomology ring of a Lie group, or the universal enveloping algebra of a Lie algebra which gives rise to the so-called quantum groups which have major physical applications.

For now the question is: can we generate a coproduct given a product, and a product given a coproduct? The answer is rather surprising. The answer is yes in both cases for finite dimensional cases, but in general one can only generate a product given a coproduct.

Then can we have a mathematical structure which has both a product and a coproduct? If such a structure exists, it is called a bi-algebra and this respects a compatibility relation where tau is transposition of the terms in the tensor product. 



Let's take the group example. Start from the upper left corner with \( g_1 \otimes g_2\) and move it horizontally:

\(g_1 \otimes g_2 \rightarrow g_1 g_2 \rightarrow g_1 g_2 \otimes g_1 g_2\)

Then take it down, across and up and see you get the same thing meaning the diagram commutes:

\(g_1 \otimes g_2 \rightarrow g_1\otimes g_1 \otimes g_2 \otimes g_2 \rightarrow g_1\otimes g_2 \otimes g_1 \otimes g_2 \rightarrow g_1 g_2 \otimes g_1 g_2\)

Usually this kind of commutative diagram are fancy ways of expressing mathematical identities. For the polynomial ring the commutativity of the diagram means that this holds:

(m+n choose k) = sum over i, j with  i+j = k of (m choose i) (n choose j)

Also Hopf algebras are special kinds of bialgebras and no wonder they have major applications in combinatorics.

Next time we'll talk about Hopf algebras. Please stay tuned.