1. Introduction and Summary
Prequantum gauge theory. Let us consider an even-dimensional manifold
X with a symplectic two-form
(
locally) as the phase space of Hamiltonian mechanics. A non-relativistic classical particle is a point in space
X moving along a trajectory in
X given by a Hamiltonian vector field
generated by a function
H (Hamiltonian) with an evolution parameter
t. In the approach of geometric quantization (see e.g., [
1,
2]), which is a neat mathematical reformulation of the canonical quantization procedure, the transition from
to quantum mechanics is carried out through the following:
The introduction of the principal U(1)-bundle over X and the associated complex line bundle with connection (we use the natural units with ) and curvature ;
The introduction of polarization on X, i.e., integrable Lagrangian subbundle of the complexified tangent bundle of X.
Both and take values in the Lie algebra . The abbreviations “" and “" here mean “vacuum" since and have no sources and define a symplectic structure on X.
The polarization makes it possible to impose on sections of the bundle the condition of independence from one “half” of the coordinates on X. The space of polarized sections of is the quantum Hilbert space and functions are the “wavefunctions" of quantum mechanics. Accordingly, the bundle with a polarization is a quantum bundle encoding the basic quantum mechanical (QM) information. By definition, a non-relativistic quantum particle is a polarized section of the complex line bundle over X.
QM as gauge theory. The above two steps (bundle , polarization ) towards the introduction of quantum mechanics do not encounter obstacles because it is simply the introduction of a bundle over X with some conditions on X and , for example, the holomorphicity condition. Moreover, it is not difficult to show that covariant derivatives in can be chosen so that they coincide with the standard operators of coordinates and momenta. Having covariant derivative acting on polarized sections of the bundle , one can construct quantum Hamiltonians as is customary in differential geometry. However, geometric quantization, like any other quantization scheme, also attempts to define a map:
which should satisfy some axioms [
1,
2]. But this does not work because of the problem with operator ordering, inconsistency with polarization and many other problems. For this reason we do not use either the map
or the metaplectic correction, but keep only the first two steps and consider quantum mechanics as the theory of fields
, which are acted upon by covariant derivatives
with canonical connection
.
Antiparticles and quantum charges. The logic of gauge theories tells us that, if there is a complex line bundle , then we should also consider the complex conjugate bundle and associate the charges and with the structure group U(1)v of these bundles, which leads to the interpretation of sections of these bundles as particles and antiparticles. Observables are introduced not through the mapping , but using a metric on X and covariant derivatives , which automatically gives the Weyl ordering.
Connection and curvature of the bundle have opposite signs compared to those in the bundle . A non-relativistic quantum antiparticle is a polarized section of the complex line bundle . In Hamiltonian mechanics, this corresponds to a point in the symplectic manifold with the evolution parameter .
The fibers of the principal bundle are circles parametrized by elements , where is a typical fiber of the associated complex line bundle with a coordinate on the fiber. A circle is a closed curve on around the point , and it has the winding number . The complex conjugate bundle has on its fibers a complex coordinate and the winding number .
In general, the winding number of a curve in
is the total number of times that the curve travels counterclockwise around a fixed point for
and clockwise for
,
For
, these curves correspond to one-dimensional representations of the group U(1) and
, defined by the homomorphisms
and
The first mapping in (
2) defines the tensor product of bundles
with
, and the second mapping in (
2) defines the tensor product of bundles
with
. We call
a
quantum charge; sections of bundles with
are
particles, and those with
are
antiparticles. Sections with
are neutral particles; these are sections of the diagonal subbundle in the bundle
. We emphasize that all quantum particles have a quantum charge, including those that have zero electric charge.
Problem statement. In classical Hamiltonian mechanics and the corresponding nonrelativistic quantum mechanics (QM), there is no concept of antiparticles. These theories describe only particles, and the number of particles is given as initial data and cannot change (particles can be created and destroyed only in quantum field theory). In Newtonian mechanics, non-relativistic particles are described by a phase space parametrized by coordinates and momenta . Antiparticles are introduced after the transition to relativistic QM with phase space , where positive-frequency solutions of free wave equations are associated with particles and negative-frequency solutions are associated with antiparticles. This interpretation is carried over to quantum field theory (QFT), where the coefficients in the expansion in positive- and negative-frequency basis solutions are replaced by operators.
In the non-relativistic limit
, positive-frequency solutions
of relativistic equations are identified with the wave functions of free non-relativistic particles with energy
, and the analogous limit of negative-frequency solutions is declared non-physical. However, antiparticles do exist at low velocities
. That is why attempts have been made repeatedly to find a transition from QFT to non-relativistic QM while preserving the concept of antiparticles (see e.g., [
3,
4] and references therein), but all these attempts were not considered successful.
Returning to Formulae (
1) and (
2), we note that differential geometry asserts that, if a particle is associated with sections
of the bundle
, then sections of the dual bundle must be associated with antiparticles (opposite charges
and
). It is well known that, if
is a rank
complex vector bundle, then there are three more complex vector bundles: the complex conjugate bundle
, the dual bundle
and the dual of the complex conjugate bundle
. For example, if quarks are described by a complex vector bundle
of rank 3 as
-columns, then antiquarks are described by the dual bundle
as
-rows. If
is a Hermitian complex vector bundle, then bundles
and
are isomorphic, as well as bundles
and
. Bundles
and
are not isomorphic and, in particular, the
k-th Chern class of
is given by
so that
.
The bundle
introduced in non-relativistic QM is a Hermitian complex line bundle [
1,
2]. Therefore, we can and must introduce the complex conjugate bundle
and its sections
, which describe antiparticles when
. In this case, the energy
E of particles and antiparticles is always the same since, in the Schrödinger equations for dual objects,
is replaced by
. Differential geometry requires distinguishing particles and antiparticles by the quantum charge
introduced above, and not by the sign of energy, and the definition in terms of bundles
is preserved for particles in any external fields and in the relativistic case. In particular, the energy of a free non-relativistic electron is positive, but, for an electron in the field of central forces generated by a proton, it is negative. Moreover, if the hydrogen atom is described by the function
, then the positron in the field of forces generated by the antiproton is described by the function
, and the experimentally observed energy levels of hydrogen and antihydrogen atoms are the same.
Using the differential geometric approach leads to the assertion that the density of conserved charges for positive- and negative-frequency complex solutions of the Klein–Gordon equation:
representing the density
of charges
associated with the bundles
and the continuity equations for the conjugated Schrödinger equations. And the interpretation of functions
as probability densities is secondary. This can be illustrated by the example of a one-dimensional harmonic oscillator, considered in this paper. Namely, the function
is the quantum charge density in both the coordinate and holomorphic Segal–Bargmann representations. However, in the holomorphic representation, the function
cannot be interpreted as a probability density (there is no
-additivity).
The inadequacy of introducing the concept of a particle and antiparticle through the sign of the eigenvalues of the operator
is obvious when considering quantum fields in curved space-time, which is discussed in detail in [
5]. In general, Killing vectors, with the help of which one could define positive-frequency solutions, do not exist at all, and this leads to the vagueness of the concept of a particle [
5]. The definition in terms of bundles
and
removes this problem since being a particle or antiparticle means having a conserved quantum charge
or
. Even in Minkowski space, the definition via the operator
does not work for a non-free particle, for example, for a relativistic oscillator. Its covariant phase space is a homogeneous Kähler space PU(3,1)/U(3), which is not fibered over either coordinate space or the momentum space, and the solutions do not contain terms with
. Instead, the oscillating particles and antiparticles are sections of the holomorphic
and, respectively, antiholomorphic
bundles over the manifold PU(3,1)/U(3) [
6].
The differential-geometric view of quantum mechanics requires another conceptual change. The “wavefunctions”
of particles and antiparticles are not functions, but sections of
vector bundles
. Therefore, they should be added not as scalars, but as vectors:
where
are bases in the fibers of vector bundles
of rank one. Also, in relativistic equations, it is necessary to separate fields with
and
as in (
4), which allows one to avoid the problem of negative energies. In particular, if
in (
4) are Dirac spinors, then the sesquilinear form on spinors is given by
where
and
and
are components of the Hermitian metric on the bundle
. By choosing
and
, we obtain a positive definite metric on the space of Dirac spinors and the corresponding Fock space. In the paper [
7], a similar change in the inner product was proposed to be obtained by changing the sign of the complex structure on the negative-frequency spinors. This corresponds to the choice
and
for opposite signs of the quantum charges
and
.
In this paper, we consider non-relativistic classical and quantum harmonic oscillator to illustrate all that has been said above. Our goal is not to say anything new about the oscillator described in textbooks. Using the example of an oscillator, we want to analyze what a particle and antiparticle with charge are at the classical and quantum levels. In addition, we focus on describing oscillators as charged particles in an Abelian gauge field with constant field strength . In other words, we regard quantum mechanics as an Abelian gauge theory with a fixed background connection with curvature on bundles .
2. Classical Harmonic Oscillator
Symplectic structure. In classical mechanics, a simple harmonic oscillator is a particle of mass
m under the influence of a restoring (attractive) force
and the equation of motion
where
is a coordinate,
,
is the frequency and
t is the evolution parameter.
Let us consider the phase space
of an oscillator with coordinate
x and momentum
p, and define on it a symplectic 2-form
with a bivector
inverse to the 2-form in (
7). We define a function (Hamiltonian) on
:
and associate with it the Hamiltonian vector field
Then, Equation (
6) can be rewritten in the Hamiltonian form
where
F is the restoring force.
Complex structure. We introduce complex coordinates on
,
with derivatives
Here,
is a length parameter defined as
so that
length
. (The dependence of all quantities on Plank’s constant
ℏ is not important for us, so we use the natural units with
)
Note that
and
form the basis of the tangent space
to
, and on it we can introduce an endomorphism
defined by formulae
It is easy to see that
where
and
. For matrix
, condition
is satisfied and
J is called a complex structure on
, so that
is a complex space with coordinate
z. From (
15), it also follows that
with the complex conjugate coordinate
, i.e., complex conjugation is equivalent to replacing
in (
16).
Harmonic oscillators. In terms of the complex coordinate
z on
, Equation (
11) and its solutions have the form
where
specifies the initial values of
x and
p. Note that
gives a trivial solution
and, therefore, the space
is usually considered as the phase space of the oscillator. We identify this solution with a particle that has a quantum charge
, coinciding with the winding number of the circle
with
in (
17). To indicate the sign of this winding number, we rewrite solution (
17) as
where
is the initial velocity.
Let us now make the substitution
in (
11). This is equivalent to replacing
and changing the sign of the complex structure on
:
and therefore obtaining a new complex coordinate
In terms of
, Equation (
11) and its solutions have the form
where
does not necessarily coincide with
. This solution has
, since it describes movement along
clockwise. Thus, the reversal of time corresponds to a change
in the sign of the complex structure, a change in the sign
and a replacement of
(particle) with
(antiparticle).
On the plane , we have
and, for
, we have
where
are initial data for
. For
, we have (
22) with
and
. Thus, we obtain
Note that exactly the same difference as between
and
exists for vortices and antivortices on the plane
; they are also associated with complex conjugate spaces and are characterized by integer winding numbers with
and
(see, e.g., [
8]).
In conclusion, we note that, in (
23), we consider a solution
with positive and
with negative frequency, as is customary in mathematics. In physics, it is often accepted the other way around, and, to do this, one simply needs to swap the complex and complex conjugate fields and solutions.
Symplectic reduction. The complex structure in (
16) can be associated with the vector field
This vector field defines the group U(1) of transformations of complex coordinates
z and
:
corresponding to rotations on the plane
with winding numbers
and
, respectively. At the same time, comparing (
10) and (
24), we see that
i.e., the Hamiltonian of the oscillator has a geometric origin. This leads to a geometric interpretation of both equations in (
26) and their solutions.
To see the geometry behind (
26), we identify the dual space
of the Lie algebra
with the generator in (
24) with the space
. Then, we can define the moment map
and the level surface
is a circle. Group (
25) preserves this circle, and the Marsden–Weinstein symplectic reduction [
9] of the space
under the action in (
25) of the group U(1) is the quotient
Thus, Equation (
26) describe the reduction of the phase space
of the oscillator to point
(moduli space). We have the map
and Solution (
18) specifies Circle (
28) lying in the fiber of the projection (
30). The antioscillator is obtained for opposite signs of the symplectic and complex structures, as in (
23). Note that the energy of both solutions
is positive and equal to
where the real parameter
can take any non-negative value.
Kähler metric. Having symplectic and complex structures on
, we can introduce the Kähler metric on
by the formula
In what follows, we use the rescaled metric
which is not dimensionless, unlike (
32).
3. Quantum Bundles
Gauge theory ⇒ QM. Repeating what was said in the introduction, we emphasize that we are not engaged in quantization in the spirit of the Dirac program [
10] and are not considering the mapping
of functions
f on phase space into quantum operators
. Instead, we consider a gauge theory on phase space described by the set
, where the connection
defines the canonical commutation relations (CCR), and the polarization
defines the Hilbert space on which the CCRs are irreducibly realized. Note that the connection
on
is given up to an automorphism of the bundle
and, by choosing different automorphisms, we can obtain coordinate, momentum, holomorphic or antiholomorphic representations that are unitarily equivalent by virtue of the Stone–von Neumann theorem. We show how all these representations are obtained from the choice of polarization
and automorphism from the group Aut
for the case of harmonic oscillators. All observables are introduced only through the operators of covariant derivatives and a metric on phase space. The field
enters in these covariant derivatives and determines the interaction of particles with vacuum.
Principal bundle . Let us consider a Newtonian particle of mass
m in one-dimensional space
. On its phase space
, the symplectic structure in (
7), the complex structure in (
12)–(
16) and the metric in (
33) are given. This particle is a harmonic oscillator if we choose the Hamiltonian of the form in (
9). This Hamiltonian defines the vector field in (
10), which has the geometric meaning of Generator (
24) of the group SO(2)≅U(1) acting on
by the rotations in (
25). A particle oscillating in space
corresponds to a point
on phase space
moving in a circle (
28) with a winding number
. Antiparticles are described by a trajectory
in
with a winding number
. The difference between particles and antiparticles is associated with orientation on circles in phase space and orientation on the time axis.
To describe quantum harmonic oscillators, it is first necessary to define a principal bundle
with structure group
and connection
. To do this, consider the space
and introduce on it a one-form
along
whose components are vector fields along
,
Here,
is a coordinate on
. Then, we introduce, on the space in (
34), vector fields
and dual one-forms
Vector fields in (
36) and one-forms in (
37) form a frame and coframe on
as a manifold.
Calculating commutator of the vector fields in (
36), we obtain the curvature
of the connection
. Note that fields interacting with
depend on
as
,
, and, for the cases of quantum charges
that we are considering here, we obtain
We call the number
the quantum charge; it distinguishes between particles (
) and antiparticles (
), with
corresponding to neutral particles. Quantum charge
is related to the winding number in (
1). In (
39), we have
for
and
for
.
Complex line bundle . The fibers of the principal bundle in (
34) over points
are groups
. Let us consider complex one-dimensional vector spaces
on which the group
acts according to the rule
. We associate with
the complex line bundles
where
. The sign “∼" means equivalence under the action of the group
on the direct product
of the space
P and
.
Spaces
are introduced as follows. Let us consider two-dimensional columns
with complex components
. These columns are acted upon by matrix
which is the generator of group
. In the space of
-vectors in (
41), we introduce a basis of eigenvectors of the matrix
J:
where “*" means complex conjugation. These vectors
are basis vectors in the spaces
, i.e.,
. Now, Vector (
41) can be expanded in
-parts:
These
are complex coordinates on fibers
of the bundles
in (
40). Note that
are complex; therefore, in the general case,
is not complex conjugate to
despite the fact that
.
We introduce a Hermitian structure on the bundles
by equipping fibers
with the Hermitian inner product:
It is obvious that Metric (
45) is invariant under the action
of the group
with
.
Complex vector bundle . Group
acts on the space of
-vectors in (
41) by multiplying on the left by the matrix
For subspaces
in
, we obtain
which coincides with the definition in (
40) of spaces
in
. Note that the action of the generator
of group
on
has the form
i.e., it is equivalent to the action of the generator
J from (
42).
The column vectors in (
44) are sections of the complex vector bundle
with the structure group given in (
46)–(
48). The
-bundle in (
49) inherits its connection
and curvature
from connections and curvature (
39) on
:
The components of this connection are given in (
35):
Accordingly, the covariant derivatives on
have the form
Note that
and, therefore, the connection
is compatible with the Hermitian metric
on
.
Operators and . We consider quantum mechanics as a gauge theory of fields
with
interacting with gauge fields
defined on these bundles. We also use the bundle in (
49) to describe
simultaneously as Sections (
44), (
47) of the bundle
.
As has been noted more than once, the spaces
of sections of bundles
are too large and they need to be narrowed down to spaces of irreducible representations of CCR by imposing conditions
where
are vector fields from the subbundles
of the complexified tangent bundle of the phase space
of oscillators. For real polarizations
, they are real subbundles of the tangent bundle
and we can consider polarization for sections of the bundle
since
.
In the two-dimensional case that we are considering, the real polarization is either the independence of sections
) from the momenta
or their independence from the coordinates
After imposing one of these conditions, we arrive at quantum mechanics in coordinate or momentum representation.
Let us see how this works for the polarization in (
55). Note that vector field
does not commute with the covariant derivatives in (
52), which is unacceptable. However, Connection (
51) can be transformed using the action of the group
of unitary automorphisms of the bundle
:
with elements
. Here,
is a real function on
. If we choose
, we obtain
Now, the covariant derivatives commute with the derivative
in (
55) and we can introduce the operators
which are the standard operators of coordinate and momentum when acting on polarized sections (
55). Thus, operators
and
are the covariant derivatives (
60) in the bundles
and the canonical commutation relation (CCR) is
This is nothing more then the curvature
multiplied by
on
.
Note that the CCR in (
61) does not depend on the choice of function
. The curvature
of the connection
on the bundle
defines both the CCR in (
61) and the uncertainty relation. Recall that
and
are vacuum gauge fields and the field
defines the potential energy of vacuum through the covariant Laplacian
along the momentum space. A more general potential energy
can be introduced either as a function of the covariant derivative
or through the component
of the metric on phase space.
To use Polarization (
56), one should apply the automorphism
, obtaining
Then, in the momentum representation, we obtain
Note that (
63) corresponds to the standard definition and (
64) reflects the fact that, for antiparticles, we have
, as discussed earlier. The CCR in (
61) does not change.
In conclusion of this section, we note that, for the automorphism generated by the element
applied to (
60), we obtain translations
and, therefore, coherent states can also be easily described within the framework of the approach under consideration.
4. Complex Polarizations
Dolbeault operators. We considered bundles
with anti-Hermitian connections
and Automorphisms (
57)–(
59), (
62) that transform covariant derivatives with this connections into operators
and
in the irreducible coordinate representation in (
60) and the momentum representation in (
62)–(
64). Now, we describe how complex polarizations
are introduced. For them, particles
are holomorphic functions of the complex coordinate
from (
12) (Segal–Bargmann representation [
11,
12,
13]) and antiparticles
are holomorphic functions of the complex coordinate
on
given in (
20) (antiholomorphic in
since
).
To define holomorphic structures in the bundles
, we introduce the Dolbeault operators
and impose the conditions
on sections
of the bundles
. These are conditions for holomorphic polarization, and their solutions are the functions
where
. Note that
and the operator
is complex conjugate to the operator
, but the functions
and
in the general case are not related by complex conjugation.
Covariant derivatives. Note that the basis vectors
of the complex line bundles
define the Hermitian Metrics (
45) and (
53) in fibers. These are Hermitian bases of Hermitian bundles. At the same time, the basis vectors
in (
68) define in
the complex bases associated with the principal bundle
having the structure group GL
, and the previously considered bundle
is a Hermitian subbundle [
14] in Bundle (
69). The function
in (
68) is an element of the group GL
that defines a mapping of Hermitian bases
into holomorphic bases
along which the holomorphic sections in (
68) of the bundles
are decomposed.
In the covariant derivatives in (
39), we initially use anti-Hermitian connections
. In complex coordinates
, we have
which implies that
Using Function (
70) as an automorphism of the bundle
, we obtain the following components of the connection in the holomorphic bases
:
Under this automorphism, the Dolbeault operators in (
66) transform into ordinary
-operators
with partial derivatives
. They commute with the covariant derivatives in (
74). Therefore, connections (
73) are holomorphic but not Hermitian.
Ladder operators. From (
71), we see that the annihilation and creation operators for bundles
have the form
when they act on sections
from (
68). After Transformations (
73)–(
75), they take the usual form
when acting on holomorphic functions
of
. The scalar product of such functions has the form
as it should be in the Segal–Bargmann representation [
11,
12,
13].
Complex real. Note that Real Polarization (
55) and Transformations (
57)–(
59) to the coordinate representation can be obtained as the limiting case of Complex Polarization (
67):
It is not difficult to verify that Operators (
76) on such functions reduce to operators
i.e., to standard ladder operators in coordinate representation. Similarly, Real Polarization (
56) and Transformations (
62)–(
64) to the momentum representation can be obtained as another limit of Complex Polarization (
67):
Operators (
76) on the functions in (
82) are reduced to ladder operators
,
in momentum representation. Here, we do not write out their explicit form.
Covariant Laplacians. Having considered real and complex polarizations, we move on to defining Hamiltonians for harmonic oscillators with
. To do this, we introduce covariant Laplacians,
acting on the polarized sections in (
68) of the bundles
. Substituting the explicit form in (
71) of covariant derivatives into (
83), we obtain
We can now introduce natural geometric Hamiltonians
defined on functions
. Here, we used the connection in (
14) between
and
.
Schrödinger equations. Recall that
and
are sections of complex conjugate bundles
. Therefore, the Schrödinger equations for them have the form
where
. Recall that, in
Section 2, we choose
and
as positive and negative frequencies to match the signs of the winding numbers. This is why, in (
86), the operator
(and not
) is used to decompose the space
into the direct sum
of positive and negative subspaces (and similarly for
) [
15]. This choice, like the distinction between particles and antiparticles, is tightly related to the choice of
orientation in spaces
and
.
Equations (
86) and (
87) can be combined into one equation for sections
of the bundle
, obtaining
Here,
is the vector field in (
24) on the base
of the bundle
. It is the generator of the group U(1)
l of rotations on
, and
J is the generator of the group U(1)
v of rotations
of the coordinates
on fibers of bundles
. This generator can be represented by a vector field or matrix (see (
48)):
Writing in the form in (
89) emphasizes and clarifies the U(1)-nature of the Schrödinger equation. Equation (
89) can also be rewritten as
where
are the winding number operators in the base
and fibers
of the bundle
. In the case that we are considering, the eigenvalues of the operator
are fixed at
, and the eigenvalues of the operator
can be any integers.
Classical and quantum: comparison. To compare Schrödinger Equation (
89) for quantum oscillator with Equation (
26) for classical oscillators, we introduce the vector
When using
Z, Equation (
26) is combined into one equation:
which can be compared with Equation (
89). Comparing the solutions of these equations, we have
where
and
are winding numbers. In addition, classical oscillator is a point moving in a circle on the plane
, and quantum oscillator is a Riemann surface
in
(or
in
), each point of which moves in a circle in
(or
),
The Riemann surface
is described in the next section.
5. Harmonic Oscillators and Orbifolds
Solutions. We discuss only solutions
of Equation (
86) in the Segal–Bargmann representation since, for
, everything is similar. We also move on to dimensionless coordinate
and omit the prime in
in the formulae below.
We have a discrete set of solutions
with the energy
(we return Planck’s constant to the expressions for the energy levels), where the term
is the energy of rotating surface
and the term
is the energy of the rotating basis vector
of fibers
of the bundle
.
Note that the squared modulus of Function (
96) is the Husimi
Q-function:
which is a quasiprobability distribution in phase space. From the point of view of gauge theory
, this function is the quantum charge density of section
of the bundle
.
Cross-sections of . Thus, we consider the bundle
with projection
:
and Solution (
96) to Equation (
86). Recall that the graph of a function
can be identified with a function
taking value in the Cartesian product,
In our case, we consider a section
of Bundle (
100),
included in Solution (
96), where the function
defines the graph in
. This graph is a one-dimensional complex surface
in
, and Solution (
96) describes a standing wave on this surface.
Orbifold . Recall that Map (
103) with
is the branched covering of degree
n, where
is the branch point. For
, the map
is the identity. Surface (
102) in
is the orbifold
, where
is the cyclic group of order
n, generated by an element
with
, i.e.,
is
n-th root of unity. Thus, we have the projection
and
is the total space of this bundle. Function (
103) is an ordinary function and its inverse
is a multivalued function
where, abusing this notation, we denote the complex coordinate on
by
.
as a cone. The orbifold is a metric cone of . Recall that the metric cone over the circle is Euclidean space and, hence, may in fact be continued non-singularly at the cone tip . For (), the metric cone is singular at the origin , since the action of is free except at the origin. The group acts on by a counterclockwise rotation through the angle about the origin, and the quotient is a cone with the cone angle :
Note that, on
in Mapping (
104), there are
n different points
,
, mapped to the same point
on
, where
. The action of group
on
defines an equivalence relation, and the part of the plane
, which is cut out by rays with angle
, is a representative of the cone
.
Metric and curvature of . Metric on
is induced from the metric on
. Using the connection in (
106) between the coordinates on
and
, we obtain
where
The Levi–Civita connection
of Metric (
107) on the cone
can be written in the form
and the Riemann curvature is
where the delta-function
indicates the singularity of curvature at the point
.
Summing up, we obtain that the eigenfunctions
of the Hamilton operator of quantum harmonic oscillator have the form in (
96) and define a fluctuating two-dimensional surface
(standing wave) in the space
with coordinate
on
, Metric (
107), Levi–Civita connection (
109) and Curvature (
110). For
, the solution
describes the rotation of the basis
in fibers of the bundle
, and this rotation with a constant frequency does not depend on
Using
, we see that, in one round around the circle in
, the circle in
is walked
n times and the basis
in fibers
rotates by
of the circle, which, in total, gives the energy of the state
. The eigenfunction
of the antiparticle has the same positive energy and opposite quantum numbers
and
, both parametrized by the fundamental group
.