Do-it-yourself: quantization of a Gauge Mechanics system

Leonardo Pedro

About this book

This book "Quantization due to time-evolution" is an ever evolving online-first book, and hopefully in the future it will evolve to be authored by the Timepiece community. It has some common features with a wiki, in the sense that it serves different kinds of readers, from the ones with a rudimentary understanding of physics to the expert readers who want to cross-check every detail of the physics theory proposed here. All these kinds of readers cannot be addressed simultaneously by a set of traditional scientific articles, where there is a single storyline.

The main storyline of this book starts in the chapter “Introduction” and it addresses readers who have the time and interest to make an effort to fully understand the physics theory proposed here.

This Prologue is an alternative storyline addressing readers (experts or not) who just want to get an idea of what the book is about, in short time and following a hands-on approach: a simple explicit system will be quantized following the methods of this book in a pedagogical way, so simple that most readers can reproduce and improve the calculations themselves. The hands-on approach will hopefully enable some of these readers to become part of the Timepiece community and authors of the book, even without reading the book in full (like it happens in a Wiki).

Nevertheless, this Prologue has many links to different parts of the book, allowing the interested readers to fully clarify the details which are not addressed in the Prologue (again, like it happens in a Wiki).

Motivation

When the Lagrangian is singular, in the classical Hamiltonian formalism there are conjugate momenta constrained to be null. This is incompatible with the canonical commutation relations of the corresponding quantum fields: the commutator of a field with its conjugate momentum cannot be null.

Another motivation for this book is that in General Relativity the diffeomorphism transformations are time-dependent transformations and many of the predictions in Gravity are produced with the Hamiltonian formalism; however the symplectic form of conservative classical mechanics is not invariant under time-dependent transformations: the most obvious way to make non-relativistic mechanics consistent with General Relativity is to formulate it as a particular field theory whose configuration space is a fibred manifold over a “time” axis [1][2], which is not much different from what we propose here..

What about the Dirac’s constrained Hamiltonian formalism? The problem is that the canonical quantization depends on the definition of Dirac brackets which requires the gauge-fixing to be both comprehensive and complete (definitions below), this is not possible in Yang-Mills theory due to the Gribov ambiguity.

Definition: gauge-fixing is comprehensive whenever it crosses all possible gauge-orbits at least once.

Definition: gauge-fixing is complete whenever it crosses all possible gauge-orbits at most once, i.e. there is no remnant gauge symmetry.

Note, the Gribov ambiguity is not solved by BRST cohomology either [3]:

“Being gauge invariant, the BFV-PI necessarily reduces to an integral over modular space, irrespective of the gauge fixing choice. Nevertheless, which domain and integration measure over modular space are thereby induced are function of the choice of gauge fixing conditions. The BFV-PI is not totally independent of the choice of gauge fixing fermion $\Psi$ .”

We propose a new formalism: the quantization due to time evolution, where gauge-fixing is complete but not necessarily comprehensive, thus the Gribov ambiguity does not necessarily hold in the Yang-Mills theory. In the following we give a brief overview of the general formalism (including links to the respective details in the book) in parallel with an explicit simple example of a gauge mechanics system.

The Hamiltonian of a classical Gauge Mechanics system

The Hilbert space is the tensor product of the symmetric and antisymmetric Fock spaces $\Gamma^s(L^2(\mathbb{R}^7\times\mathbb{Z}_2))\otimes \Gamma^a(L^2(\mathbb{R}^7\times\mathbb{Z}_2))$ [4][5]. This gives us a graded Lie superalgebra of creation and annihilation operators, of both bosonic and fermionic types. The fermionic operators correspond to a field configuration where the ghost degree of freedom is set to 1, while the bosonic operators correspond to a field configuration where the ghost degree of freedom is set to 0. The $\mathbb{R}^7$ degrees of freedom correspond to 3 fields and its corresponding time-derivatives and 1 timepiece axis.

The Hamiltonian for the gauge mechanics quantum theory in the formalism of quantization due to time-evolution has the same form as the classical Hamiltonian Action:

\begin{aligned} &H=\frac{1}{2}\int dt ... a^\dagger(t, ...)H(t, ...) a(t, ...)\\ &H(t, ...)=\frac{1}{2}\left[\pi \dot{\phi}+i\pi \lambda \phi-\frac{1}{2}|\pi|^2 -V(|\phi|^2)+\ \mathrm{h.c.}\right]\\ &\Omega=\int dt ... a^\dagger(t, ...)\left[p_\lambda \dot{\psi}^\dagger+(\pi\phi+\pi^*\phi^*)\psi^\dagger\right] a(t, ...)\\ & a(t, ...) = a(t,\phi,\dot{\phi},\phi^*,\dot{\phi^*},\lambda,\dot{\lambda},j,k)\\ & \int dt ... = \int dt d\phi d\dot{\phi} d\phi^* d\dot{\phi^*} d\lambda d\dot{\lambda}\sum_{j,k=0}^1 \\ &\{[a(t, ..., k),a^\dagger(\tau, ..., l)]\}=a(t, ..., k) a^\dagger(t, ..., l)+(-1)^{k l}a(t, ..., k) a^\dagger(t, ..., l)=\delta((t,...)-(\tau,...))\delta_{k l}\\ & \psi^\dagger\{a\}(t, ...,j, k)= a(t, ...,0,1) \delta_{j 0}\delta_{k 0}+a(t, ...,1,0)\delta_{j 1}\delta_{k 1}\\ & \psi\{a\}(t, ...,j, k)= a(t, ...,0,0) \delta_{j 0}\delta_{k 1}+a(t, ...,1,1)\delta_{j 1}\delta_{k 0}\\ & \dot{\psi}^\dagger\{a\}(t, ..., k)= a(t, ...,1,1) \delta_{j 0}\delta_{k 0}+a(t, ...,1,0)\delta_{j 0}\delta_{k 1}\\ & \dot{\psi}\{a\}(t, ..., k)= a(t, ...,0,0) \delta_{j 1}\delta_{k 1}+a(t, ...,0,1)\delta_{j 1}\delta_{k 0}\\ &[\phi,\pi]=\phi\pi-\pi \phi=i\\ &[\phi,\pi^*]=\phi\pi^*-\pi^* \phi=0\\ &[\lambda,p_\lambda]=\lambda p_\lambda-p_\lambda \lambda=i\\ &\{\psi,\psi^\dagger\}=\psi \psi^\dagger+\psi^\dagger\psi =1\\ &\{\dot{\psi},\dot{\psi}^\dagger\}=\dot{\psi} \dot{\psi}^\dagger+\dot{\psi}^\dagger\dot{\psi} =1\end{aligned}

where $(\phi,\pi)$ is a complex field/conjugate momentum field over timepiece $t$ (correspondingly). $V(|\phi|^2)$ is a functional (a polynomial or other) on $|\phi|^2$ . The BRST charge is $\Omega$ , where $\psi$ is the ghost field, $\lambda$ is the (real) gauge field and $p_\lambda$ its conjugate momentum.

The covariant derivative is $\mathcal{D}=\partial_{t}+i\lambda$ .

When defining the Hamiltonian and within each free field, we use the (symmetric) Weyl ordering [6][7] due to the fact that it conserves the exponential of operators, unlike normal-ordering [8]. This is an important property of the ordering, because we use often the Trotter product formula [9][10] (e.g. in the time-evolution operator, more about it in the following). On the other hand for the free field operators we use normal ordering, as expected in a Fock space.

A translation-invariant time-evolution

A commutative AW*-algebra is a commutative C*-algebra whose projections form a complete Boolean algebra. Conversely, any complete Boolean algebra is isomorphic to the projections of some commutative AW*-algebra [11].

Therefore, associated to the definition of probability there is always a commutative AW*-algebra. This algebra is defined for all phase-space, while the Hilbert space is only defined in the absence of constraints (or when these constraints are eliminated).

The projection corresponding to a proposition is:

\begin{aligned} &\pi(\phi(t_1)\cap ...\cap \phi(t_n))= \Psi(\phi(t_1)\cap ...\cap \phi(t_n))\Psi^\dagger(\phi(t_1)\cap ...\cap \phi(t_n))\\ &\Psi(\phi(t_1)\cap ...\cap \phi(t_n))=a^\dagger(\phi_1,t_1)...a^\dagger(\phi_n,t_n)|0>\end{aligned}

A maximal ideal is defined by the exponential vector of indicator functions and the Fock-space parametrizes the space of probability distributions, such that the probability corresponding to a maximal ideal is the modulus squared of the complex number corresponding to the maximal ideal which appears in the expansion of the wave-function in terms of exponential vectors of indicator functions. Therefore, the maximal ideal takes care of what happens sequentially along the infinite-dimensional phase-space in one elementary event, while the probability distribution takes care of what happens in parallel at different elementary events.

The above is consistent with the fact that a complete physical system is also a free system. The free field associated to a free system can be made an orthogonal(fermion)/symplectic(boson) real representation of the Poincare group depending on whether its spin is semi-integer/integer respectively, regardless of the interactions occurring within the free system [12][13]. Thus in field theory the wave-function parametrization includes a free field parametrization.

In (quantum or classical) statistical field theory, the problem we want to solve is about a probability distribution so it is about an eigenvalue problem and diagonalizing a time-evolution operator, because the eigenfunction needs not even exist and we use maximal ideals instead. On the other-hand, in classical field theory (including in numerical calculations such as the finite element method [14][15]) it is about the fields themselves and so the solution must be part of an Hilbert space (because completeness of the space is crucial for the existence proofs) and we need an alternative to the $L^2$ measure since the differential operator is unbounded with respect to the $L^2$ measure: such alternative is the Sobolev Hilbert space [16]. Thus it turns out that in general the mathematical formalism of (quantum or classical) statistical field theory may be simpler than that of classical field theory, except for particular cases.

The fact that the Hamiltonian only involves local interactions allows us to introduce an *-homomorphism where a finite number of points of the continuum space is selected. Then we can do a wave-function parametrization, which allows for a non-deterministic (infinitesimal) time-evolution for these selected points. Moreover, the (deterministic or non-deterministic) time-evolution of this finite number of points can be determined independently of the full probability distribution of the initial state, which may be a complex problem because it may involve an infinite-dimensional Sobolev phase-space with some correlation between the points due to differentiability requirements [17]. This allows us to know approximately the probability distribution of the initial and final state through numerical methods for partial differential equations and regression (Gaussian process regression [18] or statistical finite element method [19], for instance).

The fact that we are dealing with a commutative algebra is key to allow the selection of only a finite number of points of the continuum space. This is only possible because we make use of the wave-function parametrization only when it is convenient, in this case only after the selection of a finite number of points of the continuum space. We can do it because the wavefunction really is just a parametrization, without a physical counterpart.

If we had assumed that there is an infinite-dimensional canonical commutation relation algebra [20] from the beginning (as in most literature about Quantum Field Theory, instead of the commutative algebra we considered) then the *-homomorphism where a finite number of points of the continuum space is selected would not be possible. So our formalism includes the Fock-space (i.e. free fields), but not the other way around. Therefore, the Hamiltonian is quadratic in the creation/annihilation operators and no further regularization is needed: the free field parametrization can be considered a regularization by itself.

Moreover in the classical statistical field theory case where the time-evolution is deterministic1, the wave-function parametrization is crucial to define an expectation functional and its time-evolution which are mathematically well defined. Without the wave-function parametrization, the selection of only a finite number of points of the continuum space is much harder already for classical field theory [21].

The cost of the free field parametrization is that we need to implement derivatives and coordinates in continuum space as an extra structure at the local level using constraints, which allows well-defined products of fields and its derivatives at the same point in the continuum space. For an Hamiltonian which depends on the field derivatives up to first order), the commutation relations are:

\begin{aligned} &[\pi,\phi]=i\\ &[\pi,\phi(t)]=i\\ &[\pi(t),\phi]=i\\ &[\pi(t),\phi(t)]=i\\ &\pi(t=t_0)=\pi\\ %&\phi(t)=\sum\limits_{n=0}^{+\infty}\phi^{(0)}+(t-t_0)\phi^{(1)}+\phi^{(r)}(t)\\ &\phi(t=t_0)=\phi\\ &\partial_t\phi(t=t_0)=\dot{\phi}\\ %&D_t=[\partial_t,p_{(r)}(t)]\phi^{(0)}+p_{(0)}\phi^{(1)}+p_{(1)}[\partial_t,\phi^{(r)}(t)]\\ %&[iD_t-i\partial_t, H(t, ...)]=0 \end{aligned}

Note that in classical Lagrangian (and Hamiltonian) Field Theory there are also derivatives of fields (through jets of modules [22]), which are consistent with the commutation relations [23] for operator fields defined above.

This allows us to define the Hamiltonian, such that it is dependent on only a finite number of derivatives of the field $\phi(t)$ and from it derive the time-evolution operator. Note that the smooth functions are dense in the Lebesgue square-integrable space, thus the fields (and the Hamiltonian $H$ ) are assumed to be smooth.

These commutation relations imply that the field $\phi(t)$ and its canonical conjugate $p(t)$ commute with a total divergence in the Hamiltonian. Thus, the divergence will not contribute to the observable quantities for probabilities which are asymptotically constant in the continuum space 2.

The requirement of probabilities which are asymptotically constant in the continuum space still allows for spontaneous symmetry breaking, which is about the correlation of the fields at two points in space separated by an infinite distance [24].

The resulting time-evolution operator is translation-invariant. There is still no infinite-dimensional, translation-invariant and $\sigma$ -finite measure (as expected [25][26]).

We may now define Hamiltonians which are non-polynomial functions of the field $\phi(t)$ with respect to the spectral measure $d\phi(t)$ , just like it happens in Quantum Mechanics (e.g. the Hamiltonian for the Hydrogen atom). Our formalism is much more powerful and general than the peculiar notion of “continuum” through renormalization [27] also because it involves the continuum independently from renormalization and it can also be renormalized (to reproduce the standard perturbative calculations, for instance).

When defining the Hamiltonian and within each free field, we use the (symmetric) Weyl ordering [6][7] due to the fact that it conserves the exponential of operators, unlike normal-ordering [8]. This is an important property of the ordering, because we use often the Trotter product formula [9][10] (e.g. in the time-evolution operator). On the other hand for the free field operators we use normal ordering, as expected in a Fock space.

The Hamiltonian by itself does not define the theory. In the following section we will continue the the quantization due to time evolution of this system.

Constraints and gauge symmetry

In the Hamiltonian formalism, the constraints are from a technical point of view, a representation of an ideal by the zero number. By an ideal we mean an ideal in the algebraic sense.

We need to separate the ideal (gauge generator) from the gauge-invariant algebra. That is, not only the gauge-invariant algebra must commute with the ideal, but also the ideal cannot be included in the gauge-invariant algebra. This is guaranteed by non-comprehensive gauge-fixing: the gauge-invariant algebra is the sub-algebra of the commutative algebra with spectrum3 given by the fields $\phi(t),\phi^*(t),\lambda(t)$ , such that the sub-algebra commutes with the constraints. The conjugate field $p_\lambda(t)$ or its derivative $\dot{p}_\lambda(t)$ or the gauge generator are not part of the gauge-invariant algebra, since they do not commute with the corresponding field $\phi(t)$ which is included in the commutative algebra.

The ideal (gauge generator) in the gauge mechanics system is the charge operator:

\begin{aligned} Q(t)=-\dot{p}_\lambda(t)+\pi(t)\phi(t)+\pi^*(t)\phi^*(t)\end{aligned}

For consistency with General Relativity, we also impose a constraint for the observables to be translation invariant in the coordinate $t$ . As it will be discussed below, the cyclic vector defining the Hilbert space needs not be translation-invariant, just the operators corresponding to observables need to commute with the translation operator:

\begin{aligned} T(\tau)=e^{i \frac{\tau}{2} \int dt... a^\dagger(t,...) \left[p_\lambda(t)\partial_t \lambda(t)+\pi(t)\partial_t \phi(t)-i\psi^\dagger(t)\partial_t \psi(t)+\mathrm{h.c.}\right]a(t,...)} \end{aligned}

Note that the algebra of observable operators already conserves the constraints, so the Hilbert space does not need to verify the constraints. In fact, in many cases it would be impossible for the cyclic state of the Hilbert space to verify the constraints, as it was noted long ago4.

The translation in timepiece defined in Equation [eq:translation] acts as:

\begin{aligned} T(\tau) e^{i\int dt f(t) \phi(t)} T^\dagger(\tau)=e^{i\int dt f(t) \phi(t+\tau)}\end{aligned}

and for all other fields it acts in an analogous way.

Any non-trivial gauge transformation necessarily modifies the spectrum of the commutative algebra (e.g. the gauge field $\lambda(t)$ ), then one field configuration is one example of a complete non-comprehensive gauge-fixing. The gauge-fixing is non-comprehensive because the action of the gauge group on the spectrum is not transitive. Such commutative algebra has the crucial advantage that the constraints are necessarily excluded from the algebra, so that it can be used to construct a standard Hilbert space which is compatible with the constraints because the relevant operators of the commutative algebra are the ones commuting with the constraints, saving us the need to eliminate the “non-physical” degrees of freedom—setting non-abelian gauge generators to zero in the wave-function would require to solve a non-linear partial differential equation with no obvious solution for non-abelian gauge theories [28][28][28][28].

Note that it is crucial that the algebra of gauge-invariant operators is commutative. While this is not possible in the canonical quantization, it is possible with the quantization due to time-evolution as we will see later in this article. Note also that since only gauge-invariant operators are allowed, we must distinguish between the concrete manifold appearing in the phase-space and the family of manifolds (obtained from the concrete manifold through different choices of transition maps between local charts) to which the expectation values correspond.

Statistical Source Field Theory

The method of quantization described in Section [sec:time] is

inspired by the Source formalism of Schwinger [28] which is itself both an alternative to and inspired by the Feynman’s path integral, where time-ordering [28][28] plays a key role.

We here use the term field meaning a function of a parameter $t$ which we call timepiece. The timepiece has a close relation with the time, the main difference is that our fields are part of the phase-space of the theory: the state of the system is defined by functions of the timepiece $t$ . Thus the phase-space has some similarities (and many differences) to the space of trajectories in time of the Lagrangian formalism. Then, the time-evolution will modify the state of the system as a function of the time parameter $\tau$ .

The timepiece $t$ from the phase-space and the time $\tau$ from the time-evolution are deeply related, although they play different roles in our framework in the cases where the time-evolution $U$ has a non-trivial gauge symmetry.

Our fields are best described as source fields and we are dealing with a statistical source field theory. Using a wave-function, we can parametrize the probability distribution for a source field in timepiece. The linear space generated by all wave-functions is the symmetric Fock space $\Gamma^s(L^2(\mathbb{R}))$ [28].

The Fock space has the properties of a continuous tensor product of Fock-spaces corresponding to fields defined in infinitesimal timepiece-intervals, i.e. $\phi(t)dt$ . The time-evolution will not only advance the timepiece-intervals forward, but it will modify the wave functions corresponding to each timepiece interval accordingly to an Hamiltonian which plays here the role of the connection in a covariant derivative. With abuse of language, we can describe the situation as a continuous tensor product of initial-value (i.e. Cauchy) problems, instead of just one initial value problem as in standard Quantum Mechanics.

The timepiece $t$ from the phase-space and the time $\tau$ from the time-evolution are deeply related, although they play different roles in our framework in the cases where the time-evolution $U$ has a non-trivial gauge symmetry.

Due to the BRST cohomology, we can add to the Hamiltonian the term:

\begin{aligned} \{\Omega,i\psi\lambda\}=\lambda\dot{p_\lambda} -i\lambda(\pi\phi+\pi^*\phi^*)-i\psi^\dagger\dot{\psi}\end{aligned}

We obtain a BRST invariant Hamiltonian in the Weyl gauge, where the translation in timepiece can be factorized, allowing us to redefine a new Hamiltonian $H_3(t)$ integrating 3D space only and without explicit dependence in the timepiece $t$ :

\begin{aligned} &H_W=H+\{\Omega,\int dt... a^\dagger(t,...)\psi(t) \lambda(t)a(t,...)\}=\int dt... a^\dagger(t,...)\left[-i\psi^\dagger\dot{\psi}-p_\lambda\dot{\lambda} +\pi\dot{\phi}+\pi\dot{\phi}-\frac{1}{2}|\pi|^2 -V(|\phi|^2)\right]a(t,...)\\ &=T(\tau) e^{i \frac{\tau}{2} dt \int H_3(t)}\end{aligned}

Considering only operators respecting time-ordering, we can move all operators to a single slice of time and then the predictions coincide with the canonical (Dirac) quantization of the gauge mechanics classical theory.

The time-evolution of $\phi$ decouples from the time-evolution of $\lambda$ .

Thus, the gauge field $\lambda$ can be studied first, without taking into account the complex field $\phi$ . We do this by integrating only the $\lambda$ field, while keeping the $\phi,\phi^*$ fields as external variables to the theory and within the theory it is as if we would only consider operators which do not depend on $\phi,\phi^*$ . Then, we are left with the $\lambda$ field only, which commutes with $H_3(t)$ and also with the remnant gauge transformations, then the corresponding gauge theory is that of a gauge-invariant field which is constant in time, which has an easy solution.

After integrating the $\lambda$ field, we obtain operators which depend on $\phi,phi^*$ only and do not depend on $\lambda$ . Note that we could also assume that the operators do not depend on $\lambda$ because they were chosen not to depend on

$\lambda$ (due to an argument related to gauge-invariance as we will see below) and not because the fields

$\lambda$ were integrated first. This is not incompatible with our results because the remnant-gauge-invariant operators involving $A_{i a}$ only, are also fully gauge-invariant. Thus our framework is more general but it is compatible with reference [28] because we can also simply choose remnant-gauge-invariant operators involving $A_{i a}$ only, which will be automatically fully gauge-invariant even if we just ignore the $A_{0 a}$ fields.

Considering the remnant gauge-symmetry and only a single slice of time, if we consider the Hamiltonian $H$ and time-evolution $U$ defined as:

\begin{aligned} H&=\int dt\ (p^2(t)+V(x(t),t))\\ U(\tau)&=T(\tau) e^{i \frac{\tau}{2} H}\end{aligned}

Where $V(x(t),t)$ is a potential dependent on the position operator and possibly also timepiece-dependent.

We will use now the Trotter exponential product approximation [28][28], verifying for small $\epsilon$ and $A$ , $B$ , $A+B$ self-adjoint:

\begin{aligned} e^{i\epsilon A}e^{i\epsilon B}= e^{i\epsilon (A+B)-\frac{\epsilon^2}{2}[A,B]+i\mathcal{O}(\epsilon^3)}\end{aligned}

This is a good approximation since it works for unbounded self-adjoint operators $A,B$ .

Then the time evolution is:

\begin{aligned} U(\tau)&=e^{i \int_0^\tau dz \int dt\ p^2(t)+V(x(t),t-z) } T(\tau)\end{aligned}

Where the exponential above stands for the time-ordered (with parameter $z$ ) product. Thus the Fock-space parametrization of a statistical field theory allows us to implement the concept of time-ordering [28][28] consistently.

If we relax the mathematical rigor for a moment and imagine a source field completely localized in one instant of the timepiece $t$ , then the time-evolution (with time $\tau$ ) of that source field could be described as a physical field function of time $\tau+t$ with initial conditions defined at time $t$ .

In fact, a statistical source field theory can be seen as the solution of a particular Schrödinger equation. In this particular case, the Hamiltonian is given by the generator of the time-evolution $U$ . Such generator is not time-dependent, however it does depend on the timepiece $t$ but such parameter is part of the Fock-space which is a particular case of the Hilbert space appearing in the Schrödinger equation.

This gives us a big freedom to choose the cyclic vector defining the Hilbert space (which is related with the initial state of the system).

Note that for self-adjoint operators with null-expectation value, the contributions coming from the timepieces $t$ corresponding to vaccuum expectation values are null. Since the translations in timepiece $t$ of the complete system conserve the Hamiltonian and the constraints equations, then setting the generator of translations to zero in no way conflicts with the constrained Hamiltonian system.