Download On Gauge Invariance and Covariant Derivatives in Metric Spaces

Gauge Invariance and Covariant Derivatives in Metric Spaces
Kaushik Ghosh∗
Vivekananda College,University of Calcutta, 269,
Diamond Harbour Road, Kolkata - 700063, India
arXiv:1702.02384v6 [physics.gen-ph] 3 May 2017
Abstract
In this manuscript, we will discuss the construction of a covariant derivative operator in a quantum
theory of gravity. We will find that we have to use connection coefficients more general than the LeviCivita symbols in a general quantum theory of gravity irrespective of coupling with gauge and matter
fields. We will show this for a canonical quantum theory of gravity. We will also find that quantum
fluctuations in geometry give additional degrees of freedom associated with the generalized connections.
These additional degrees of freedom vanish in the classical spacetime manifolds that are solutions of the
Einstein equations with connections given by the Levi-Civita symbols. These additional fields also arise in a
path integral formulation of quantum gravity with the Palatini action. The degrees of freedom associated with
the generalized connections can be described by the torsion tensor and a symmetric second rank covariant
tensor together with its ordinary partial derivatives. The later field is not considered in quantum theories of
gravity. This field and torsion may become significant in quantum cosmology. We will conclude this article
with an alternate proof of the Poisson equation. We will find that the self energy of a point charge can be
zero in the potential approach.
Key-words: connections, quantum gravity, canonical quantization, Palatini action, inflation, Poisson equation
MSC: 83C45, 83C47, 58A05
PACS: 04.60.Bc, 04.20.Fy, 02.40.Hw
∗
E-mail ghosh [email protected]
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I.
INTRODUCTION
In this article, we will discuss a few aspects of geometry relevant to a quantum theory of gravity. We
will also give an alternate proof of the Poisson equation in electrostatics. In Sec.II, we will discuss the
issue of construction of a covariant derivative operator in a quantum theory of gravity. We can introduce a
metric to a curved spacetime, provided it satisfies a few very general topological conditions [1]. To construct
differential equations for different fields, we have to introduce additional structures into the spacetime.
These structures are known as connection coefficients. They can be introduced in a differential manifold
independent of metric [2]. We can construct covariant derivatives in a manifold in terms of ordinary partial
derivatives and connection coefficients. Covariant derivatives have three general properties which are related
to the corresponding properties of ordinary derivatives [2]. They are also assumed to possess two additional
properties [2,3]. We will mention these properties in Sec.II. In this article, we will find that we have to use
connection coefficients more general than the Levi-Civita symbols in a general quantum theory of gravity
irrespective of coupling with gauge and matter fields. We will use a canonical formulation of quantum gravity
to show this. This is consistent with a path integral formulation of quantum gravity with the Palatini action.
These connections may be asymmetric in the lower indices and need not to be compatible with metric or
any symmetric second rank covariant tensor [2]. Formulation of differential geometry of a manifold with a
general set of connections has been discussed by Lovelock and Rund [2]. This geometry will be important to
formulate a quantum theory of gravity which uses connections more general than the Levi-Civita symbols.
Generalized connections give additional fields even when gravity is not coupled with gauge or matter fields.
These are vanishing in the classical spacetime. This is characteristic of the Palatini action. These fields are
necessary in a canonical quantization scheme. The antisymmetric part of generalized connections can be
described by the torsion tensor. We will find from Eqs.(11,12) of the following section, that the symmetric
fluctuations of connections off the Levi-Civita symbols can be described by a symmetric second rank covariant
tensor and its ordinary partial derivatives. Thus, a general quantum theory of gravity can be described by
metric, a symmetric covariant second rank tensor and the torsion tensor. The second field is usually not
considered in quantum theories of gravity. These two fields may become relevant to explain the quantum
cosmological problems like inflation. Note that, here we are introducing these two fields as elementary fields
irrespective of coupling with gauge or matter fields. They are similar to metric in this respect. Coupling of
gravity with fermions may introduce torsion in a spacetime [4]. In this case torsion will be non-vanishing in
the semi-classical limit of quantum fields in a curved spacetime. In this article, we have not considered the
dynamics of a quantum theory of gravity. This an involved problem due to presence of non-trivial constraints
[3,4,5]. We have only considered a general mathematical issue which will be there, even in a general free
theory of quantum gravity which is not a quantum field theory in a fixed background, provided there exist
dynamical variables. It is a non-trivial issue to extend the Maxwell equations when connections are not
given by the metric compatible Levi-Civita symbols. This is important for the strict current conservation
law and potential description. The fields associated with the generalized connections can also be related to
geometric coupling between gravity and gauge fields. We will discuss these aspects in the next section.
In Sec.III, we will discuss an alternate proof of the Poisson equation. We have mentioned this proof in
reference [6]. Here, we will clarify and illustrate it further. We will find that we can apply the Gauss’ integral
theorem to vector fields which are divergent of degree less than three. This is significant, since we can not
directly apply the methods of analysis to singular functions. We will elucidate this aspect in the proof. We
will find that the proof indicates that self energy of a point particle is zero in the potential approach. This
is different from the conventional expressions, where we do not show much concern to the applicability of
the Gauss’ integral law to divergent fields [7]. This also follows directly from the definition of electrostatic
energy in terms of the electric potential. This is physically expected. In absence of any interaction, it is
welcome that the reference value of energy of an isolated system is finite and zero. However, we can directly
start from a Lagrangian written in terms of the fields. This will lead to a divergent value for the self-energy
of point charge [7,8]. These discussions show that the construction of the stress-tensor of electromagnetic
fields is a non-trivial issue when point charges are present. This article removes an error in Sec.3 of reference
[6]. The Maxwell equations remain gauge invariant in a spacetime where a covariant derivative obeys the
conditions mentioned at the beginning of the following section.
II.
QUANTUM GRAVITY AND COVARIANT DERIVATIVES
In this section, we will address the issue of construction of a covariant derivative operator in a quantum
theory of gravity. We will find that we have to use connections more general than the Levy-Civita symbols
in a general quantum theory of gravity irrespective of coupling with gauge and matter fields. This aspect is
3
transparent in a canonical quantization scheme of gravity. Generalized connections give additional degrees
of freedom that can be described by the torsion tensor and a symmetric covariant second rank tensor. This
may be important for quantum cosmology.
Three properties of ordinary derivatives, and transformation rules of tensors under change of coordinate
systems are used to construct covariant derivatives in a general manifold [2]. Consequently, covariant derivatives in a curved spacetime inherit these properties. We state them in the following:
(I) The linearity property:
..αi ..
..αi ..
..αi ..
..αi ..
∇µ [aM..β
+ bN..β
] = a∇µ M..β
+ b∇µ N..β
j ..
j ..
j ..
j ..
(1)
..αi ..
..αi ..
where, a, b are two constants and M..β
, N..β
are two well-behaved tensor fields.
j ..
j ..
(II) For a well-behaved scalar field f , and a vector field tµ , t(f ) = tµ ∇µ (f ). Here, t(f ) denotes directional
derivative of the scalar field in the direction of the vector field.
(III) The Leibnitz rule:
..αi .. ..αi ..
..αi ..
..αi ..
..αi ..
..αi ..
∇µ [M..β
N..βj .. ] = [∇µ M..β
]N..β
+ M..β
[∇µ N..β
]
j ..
j ..
j ..
j ..
j ..
(2)
..αi ..
..αi ..
here, M..β
, N..β
are two arbitrary well-behaved tensor fields.
j ..
j ..
Apart from the above three rules, covariant derivatives, in curved manifolds that are relevant to classical
general relativity, also satisfy the following two conditions.
(IV) We define the torsion tensor through the following relation:
α
[∇µ ∇ν − ∇ν ∇µ ]f = Tµν
∇α f
(3)
α
where, f is a well-behaved scalar field. Tµν
is the torsion tensor. In classical general relativity, the torsion
tensor is assumed to be zero.
(V) Commutativity between contraction and covariant derivative:
..αi ..
..αi ..
∇µ [C(M..β
)] = C(∇µ [M..β
])
j ..
j ..
(4)
here, C( ) denotes contraction operation.
We can construct a covariant derivative operator having properties (I - III):
α
∇µ Aν = ∇′µ Aν − Cµν
Aα
(5)
α
Here, Cµν
are connection coefficients. In classical theory of general relativity, we usually choose ∇′µ ≡ ∂µ ,
where ∂µ is the ordinary partial derivative. The corresponding connections are known as the Chistoffel
symbols [3]. We can introduce additional conditions on the Chistoffel symbols. In particular, we introduce
the Levi-Civita symbols through the following metric compatibility conditions [3]:
∇µ [gαβ ] = 0
(6)
We have the following unique expressions for them:
α
Cµν
= Γα
µν =
1 ακ
g [∂µ (gκν ) + ∂ν (gµκ ) − ∂κ (gµν )]
2
(7)
The above expression shows that the Levi-Civita symbols are dependent on the derivatives of metric and
are symmetric in the lower indices. The corresponding torsion tensor is zero. In classical general relativity,
the right hand side of Eq.(3) is replaced by zero. The familiar solutions of the Einstein equations satisfy the
torsion-free conditions [3]. In the following, we will show that we have to use connections more general than
the Levi-Civita symbols in a canonical quantization scheme of gravity.
We first consider a path integral formulation of quantum gravity with the Palatini action to describe
gravity. The action is given by [3]:
4
S=
Z
√
(d4 x) −g[R]
(8)
Where, R is the scalar curvature. In the Palatini formulation, connection coefficients and metric are considered as independent variables. We have the following expression for the covariant derivative operator:
α
∇µ Aν = ∇′ µ Aν − Cµν
Aα
(9)
α
Here, Cµν
is a set of arbitrary connection coefficients. They can be symmetric or asymmetric in the lower
indices. Here, ∇′ µ is a fixed covariant derivative, which have all the five properties mentioned before. In
particular the torsion tensor is zero for ∇′ µ . We choose ∇′ µ to be given by the following expression:
∇′ µ Aν = ∂µ Aν − Γα
µν Aα
(10)
Here, Γα
µν are the Levi-Civita connections and are symmetric in the lower indices. The symmetric part of
the general connections in Eq.(11) can be given by the following expression [2]:
α
α
Sµν
= Γα
µν + C(µν)
1
= b̃ακ [∂µ (bκν ) + ∂ν (bµκ ) − ∂κ (bµν )]
2
(11)
Where, Γα
µν are the Levi-Civita symbols and given by Eqs.(6,7). bµν is a non-singular symmetric covariant
tensor and can be expressed as:
bµν = gµν + aµν
(12)
The inverse of bµν , b̃µν , is a contravariant tensor [2] and can be expressed as b̃µν = g µν + dµν . Note that
α
b̃µν is different from bµν and Sµν
satisfy the compatibility conditions: bµν|α = 0, where the bar denotes
α
α
covariant derivative with connections Sµν
. Thus, the symmetric part of Cµν
can be described by aµν , gµν
and the ordinary partial derivatives of both. We can easily generalize Eq.(12) by making aµν dependent on
gµν , although the symmetric deviations of connections away from the Levi-Civita symbols can always be
described by a symmetric second rank covariant tensor and its partial derivatives. The antisymmetric part
α
α
of Cµν
is given by the torsion tensor. In the Palatini action principle, metric and connections Cµν
are taken
′
to be independent variables, while ∇µ is held fixed [3]. We have the following expression for the Riemann
curvature tensor [3,9]:
κ
κ
Rµνακ = R′ µνα + 2∇′ [ν Cµ]α
+2
X
λ
κ
[C[µ|α|
C|λ|ν]
]
(13)
κ
Here, R′ µνα is the Riemann curvature tensor associated with the derivative ∇′µ in Eq.(9), and is given by
the familiar expression in terms of ordinary partial derivatives of Γα
µν [2]. In the following, we will find that
the Palatini formulation is consistent with a general quantum theory of gravity.
We now consider a general formulation of quantum gravity using the canonical quantization method. In a
canonical quantization scheme, all the field variables including the electromagnetic fields, potentials, metric
become operators on a Hilbert space. We represent these operators by carets. Connections act on the tensor
indices and we represent them by the symbols: Θ̌α
µν . Connections may contain components of metric and
their time derivatives. In a Hamiltonian formulation, gµν is replaced by: hµν = gµν + nµ nν , where nµ is
the unit normal to constant ’time’ surfaces [3]. Here, the metric signature is taken to be − + ++. The
corresponding conjugate momenta are given in [3]. We use the symbols ĥ, π̂ to denote the corresponding
collection of operators. The Levi-Civita symbols contain (ĥ, π̂). We now express covariant derivative operator
in the following form:
ˇ µ − Θ̌α (ĥ, π̂)]Âα
ˇ µ Âν = [I∂
∇
µν
(14)
5
ˇ
Here, Θα
µν are the Chistoffel symbols which need not to be the same as the Levi-Civita symbols [3]. I is the
ˇ µ q ν , where q µ is
identity operator on the spatial and tensorial indices. We now consider the operator: q µ ∇
a vector field. This operator contains canonical conjugate pairs of variables if we choose the connections to
be given by the Levi-Civita symbols. In this case, we will have the following expression:
ˇ µ q ν ] |Ψi 6= 0
[q µ ∇
(15)
valid for an arbitrary state |Ψi. We can not have a class of states for which the expectation value of the
operator in the l.h.s is zero with negligible fluctuations for all well-behaved vector fields. This may be valid
only in the classical limit, and is a subject similar to the familiar Ehrenfest’s theorems in non-relativistic
quantum mechanics. Similar discussions will remain valid even if we choose connections to be given by
the operator versions of Eqs.(9,10). In general, connections will contain canonical pairs of variables from
the gravity sector to have proper classical limit of the Levi-Civita connections, and the concept of parallel
transport do not remain well-defined in an arbitrary quantum gravitational state. Thus, the notion of
parallel transport is not well-defined in a quantum theory of gravity. This is expected. We now consider
the metric compatibility condition. The metric compatibility condition given by Eq.(6) is to be replaced by
ˇ µ [ĝαβ ] ≡ 0. The action of ∇
ˇ µ [ĝαβ ] on any state is zero if the connection operators
the operator identity: ∇
are given by the Levi-Civita connection operators and we choose a particular operator ordering. Here, we
always keep metric operator as the successor in any index changing operation. This agrees with the identity:
ˇ µ [ĝαβ ] ≡ 0. However, the operator version of the metric compatibility condition is contradictory with a
∇
canonical quantization condition. As mentioned above, in a Hamiltonian formulation, gµν is replaced by:
hµν = gµν + nµ nν , where nµ is the unit normal to constant ’time’ surfaces [3]. We regard the covariant
components of nµ as non-trivial operators since: nµ = gµκ nκ . We have the following algebraic relation
essential in a quantum theory of gravity provided ĥαβ is a dynamical variable:
µν
[ĥαβ (x), π̂ µν (y)] = [ĝαβ (x) + ĝακ (x)ĝβτ (x)nκ (x)nτ (x), π̂ µν (y)] = Xαβ
[ĝ, (x, y)]
(16)
Here, the right hand side is a singular distribution of (x, y) with appropriate symmetry in the tensorial
indices. It is appropriate to consider the above equation as an algebraic relation essential in a quantum
theory of gravity instead of worrying about causal interpretation. We have seen before that the notion of
parallel transport is not well-defined in a full quantum theory of gravity. On the other hand, Eq.(16) remains
valid when we quantize small metric fluctuations around a fixed background with ĥαβ in the l.h.s regarded as
quantum perturbations to the fixed background and ĝ in the r.h.s is replaced by the fixed background metric.
In this case, we need not to worry about the middle term in Eq.(16). In a Hamiltonian formulation with
the Einstein-Hilbert action, we have non-trivial constraints and it is not possible to separate the dynamical
variables completely from the auxiliary variables [3,5]. However, with a particular choice of constant ’time’
surfaces, we can assume that there exist a coordinate system so that at least one of the space-space component
of metric, g33 , is a dynamical variable in a neighborhood of a point ′ x′ with g33 (x) = h33 (x). The x0 -direction
of the coordinate system is chosen to be coincident with nµ and the x3 -coordinate curve is chosen to be
embedded in the constant ’time’ surfaces in this neighborhood. We then have,
33
[ĝ33 (x), π̂ 33 (y)] = X33
[ĝ, (x, y)]
(17)
ˇ xκ on both side
Where, the point ′ y ′ belongs to the above mentioned neighborhood of ′ x′ . We can apply ∇
of the above equation. The right hand side remains non-vanishing in general. This is the situation with
quantum field theories of gauge and matter fields in a fixed background [10]. The left hand side vanishes
if we impose the operator versions of the metric compatibility conditions. This is a contradiction. Thus,
it is not possible to use the Levi-Civita symbols as connections in the above mentioned coordinate system.
Similar situation will remain valid for any point in the manifold where we can introduce the constant ’time’
surfaces. The above contradiction will also arise with a different choice of constant ’time’ surfaces. Thus,
there will be a multitude of coordinate systems where we can not use the Levi-Civita symbols as connection
coefficients. Thus we find that, in a canonical quantization scheme, it is necessary to extend the Levi-Civita
connections. Note that, we have not considered the dynamics of a quantum theory of gravity which will
require separating dynamical variables from auxiliary variables. This an involved problem [3,4,5]. We have
only considered a general mathematical issue which will be there, even in a general free theory of quantum
gravity which is not a quantum field theory in a fixed background, provided there exist dynamical variables.
ˇµ
We may use Eqs.(9,10) to define connections so as to have a proper classical limit and the operators ∇
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will use the operator versions of these generalized connections. We have to consider additional degrees of
α
freedom when we use Eqs.(9,10) to define connections. This is because the connections Cµν
in Eq.(9) are
independent fields. This is similar to the path integral formulation with the Palatini action discussed before.
We will later find that these degrees of freedom may be useful in the cosmological problems. Thus, in a
general formulation of canonical quantum theory of gravity, we will have to remove the metric compatibility
conditions and use connections more general than the Levi-Civita symbols. We also find that the Palatini
formulation is more appropriate to construct a path integral theory of quantum gravity than the theories
that use metric as the only independent variables. The situation with string theory is different. Finally, we
can symmetrize the different canonically conjugate operators present in Θ̌α
µν to make connection operators
Hermitian. In this case, the operator version of the metric compatibility condition will not remain exactly
valid even for the Levi-Civita symbols.
We now briefly consider electrodynamics in a curved space with a general set of connections discussed
before. This is a non-trivial issue since the connections do not satisfy metric compatibility conditions. We
assume the electromagnetic field tensor: Fµν , to be antisymmetric. We have the following generalization
of the strict current conservation law, if we assume the conventional inhomogeneous Maxwell equations:
∇ν F νµ = −j µ , continue to remain valid with a general set of connections:
1 α
∇µ j µ = −Rµν F µν − Tµν
∇α F µν
2
(18)
α
Here, Rµν is the Ricci tensor which is not symmetric with a general set of connections and Tµν
is the torsion
tensor [2]. It is an important problem to interpret this relation. This relation indicates that geometry can
influence measurement of electric charge. We can assume that the homogeneous Maxwell equations continue
to hold with a general set of connections. In this case, it may not be possible to describe the fields by the
potentials even if we introduce torsion in the definition of fields in terms of the potentials. Alternatively, we
may continue to assume that the field tensor is described by a set of potentials in a gauge invariant way:
F αβ = g αµ g βν Fµν
= g
αµ βν
g
[∇µ Aν − ∇ν Aµ −
(19)
α
Tµν
Aα ]
This is consistent in the Palatini formulation, since the potentials are analogous to the connections in the
geometric theory of gauge fields [11]. The fields are invariant under gauge transformations: Aµ → Aµ + ∂µ χ.
It is easy to show that the electromagnetic field tensor will not satisfy the homogeneous equations [2].
Instead, we have the following three form field:
∇[µ Fνα] = Bµνα
(20)
Note that, we have used the contravariant field tensor to write the inhomogeneous equations coupled with
the sources while the covariant field tensor to introduce the potentials and the B fields. This is important
to use the definitions of Riemann curvature tensor and torsion tensor in absence of the metric compatibility
conditions. The B fields arise from geometric coupling between gravity and gauge fields, and are purely
quantum gravitational in origin. In four dimensions, there will be four components of this electromagnetic
B field. Similar fields will appear for the non-Abelian gauge theories. These fields may become significant
in a complete quantum theory of gravity coupled with gauge and matter fields. These fields will vanish in a
classical spacetime with the Levi-Civita symbols as connection coefficients.
α
The Cµν
fields of Eqs.(9,10) can be important to solve some fundamental cosmological problems like the
inflation problem. Their effects are significant at the Planck scale. These fields will vanish for manifolds
that are solutions of the Einstein equations with connections given by the Levy-Civita symbols. Thus,
the classical cosmological models, where the connections are given by the Levi-Civita symbols and obey
the metric compatibility conditions, can be taken as the vacuum for these fields. These fields can not be
detected by fluctuations in metric. The torsion tensor can be used to find antisymmetric fluctuations in
connections away from the Levi-Civita symbols. We have introduced a symmetric covariant second rank
tensor to describe symmetric fluctuations in connections away from the Levi-Civita symbols [2]. This is
evident from Eqs.(9,10,11,12). Thus, the degrees of freedom associated with the generalized connections can
be described by the torsion tensor and a symmetric second rank covariant tensor together with its partial
drivatives. The Palatini action can be used to study the free quantum theory. We can generalize the Palatini
action by including the gauge and matter field actions to describe the interacting theory. We can consider a
perturbation theory by considering the torsion tensor to be small compared to the Levi-Civita symbols and
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the aµν fields in Eq.(12) to be small compared to gµν . Note that, the torsion may not vanish in the classical
limit of quantum fields in curved spacetime when we have fermions present as matter fields [4].
We conclude the present article with the following discussions within the context of quantum fields in
curved spaces. If we accept the Big bang model of the universe, time reversal symmetry may not be much
meaningful in the very early universe. To illustrate, in nonrelativistic quantum mechanics time reversal
symmetry in a time independent potential means: u∗ (~r, −t) = u(~r, t), where u(~r, t) is the solution of the
Schrodinger equation [12]. Thus, we have to be able to define the negative time states: u(~r, −t) which is not
obvious if we take the Big bang singularity to be the beginning of time [3]. We can try to solve the problem
by extending time through the initial singularity and by considering it to be the Big crunch singularity of
another universe. On the other hand, we can consider the formation of a black hole through gravitational
collapse and it’s subsequent evolution by Hawking radiation. We can easily consider a system in a pure
state collapsed to a black hole and subsequently evolved into thermal radiation [13]. If we can assemble the
thermal radiation to collapse to a black hole, the black hole will subsequently evolve into a mixed state and
not into the initial pure state we began with. This is not time reversal symmetric. The presently observed
anisotropy in the cosmological microwave radiation indicates that parity was only an approximate symmetry
in the early universe [14]. Thus, it may be so, that charge conjugation symmetry were broken in the early
universe. This may be useful to explain the observed particle-antiparticle asymmetry. There will be other
issues like the spin-statistics theorem.
III.
A PROOF OF THE POISSON EQUATION
In this section, we will consider an improved derivation of the Poisson equation for the electric field of
a static source [3]. We first note that, we can not apply the integral version of the divergence theorem if
the vector field is not sufficiently
well-behaved at any point inside the region of interest. In one dimension,
Rx
we can use the relation: x12 (df /dx)dx = f (x2 ) − f (x1 ), only if f (x) is sufficiently well-behaved for all x
x
between and including the limits. The expression Ex (∆x) = Ex (r = 0) + ∆x[ dE
dx ]r=0 is not well-defined for
a point charge at the origin. As we will find from the following discussions, in the case of an inverse-square
field, we may replace Ex (r = 0) by the value of Ex at any other point within the neighborhood containing
~ E.
~ However, this will give vanishing result for the volume integral of
the origin in the volume integral of ∇.
2
~
divergence of E. This will violate Gauss’ integral law. This is due to the fact that ( rr2 ) is not well-defined
at r = 0, and we can not replace it by 1 at r = 0 which is the essential lower limit of the radial integration
to include the point charge. Nevertheless, in the following section, we will show that in the spherical polar
coordinates, an explicit volume integration of the divergence of the electric field of a point charge gives us
the total charge when the charge is at the origin. Also for a point charge at the origin, the volume integral
of the divergence of the electric field is vanishing when the volume of integration does not include the origin.
This together with the fact that the total flux of the electric field of a point charge at the origin over an S 2
centred at the origin is given by ǫq0 , lead us to a relation for the electric field of a point charge analogous to
the integral version of the divergence theorem. We can generalize this relation for a point charge not at the
origin and have the Poisson equation for the electrostatic field.
We first consider a point charge at a radius vector ~a on the Z -axis. In the following discussions we will
ultimately consider the point charge to be at the origin. We use the inverse square law in the spherical polar
coordinates, for the electric field of a point charge at the position vector ~a. The electric field is azimuthally
symmetric and the corresponding component is zero. We integrate the divergence of the electric field over
a sphere centred at the origin. The radius vector of the point charge is ~a, and the radius of the sphere is
greater than |~a|.
We first consider the integral of the radial derivative term:
4πǫ0 Ir =
Z
dv[
~
1 ∂ r2 R.r̂
)]
(
2
r ∂r R3
(21)
~ = ~r − ~a, and the radius of the volume of integration is greater than |~a|. We considered ~a to be on
Here R
the Z-axis. The measure in the integral contains a (sin θ) term in the spherical polar coordinate system.
Hence, the complete triple integrand is divergent of degree two at the point charge. Thus, the integral is
well-behaved and the Riemann sum remains independent of our choice of the point of evaluation of the
integrand. In particular, we can evaluate the integrand in the Riemann integral infinitesimally away from
the position of the point charge, and use the rule of the total integrals to the above radial integral to have,
8
4πǫ0 Ir =
Z
r3 q
dΩ 2
2
[r + a − 2racos θ]3/2
Z
r2 q(~a.r̂)
− dΩ 2
[r + a2 − 2racos θ]3/2
(22)
In the limit a = 0, the above expression gives the result Ir = ǫq0 . We now consider the integral of the polar
angular derivative term. This integral is also a total integral in the polar angle θ. The upper limit of the
integral gives vanishing contribution. The integral can take finite value only from the lower limit θ = 0 part.
This term and hence the value of the corresponding integral is given by the following expression:
4πǫ0 Iθ = −[2π
Z
r′
rdrsin (δθ)(
0
~ θ̂
R.
)]
R3
(23)
Here r′ > a. In the limit δθ → 0 and a → 0, the above integral vanishes. There isR no azimuthal component
~ ~r .Edv
~
of the electric field when the point charge is on the polar axis. Hence, we have ∇
= ǫq0 , when the
point charge is at the origin.
R
~ ~r .Edv
~
is vanishing if the integrating volume does not
When the point charge is at the origin 4πǫ0 ∇
include the origin. If we consider any annular region centred at the point charge at the origin, the volume
~ ~r .E)
~ is given by the following expression:
integral of (∇
Z
qdΩ[(
r2 3
r1
) − ( )3 ] = 0
r2
r1
(24)
~ ~r .E)
~ is vanishing for any arbitrary volume not including the
It is easy to show that, the volume integral of (∇
point charge at the origin. Thus, we have the well-known Poisson equation for a point charge at the origin,
~ ~r .( r̂ ) = 4πδ 3 (~r)
∇
r2
(25)
The above discussions together with the fact that the total flux of the electric field over a closed surface is
Q
ǫ0 , allows us to write the corresponding integral version of Gauss’ divergence theorem for the electric field of
a point charge at the origin. We can derive a corresponding expression for the divergence of the electric field
~ = (~r − ~r′ )
of a point charge not at the origin. We follow the above derivation with ~r and ~a replaced by R
′′
′
′
~
and a = (~r − ~r ) respectively, and consider the explicit volume integrals over a sphere centred at ~r′ . Here
~r′ is the position of the point charge. We consider the inverse square law to find the flux of the electric field,
and we have the Poisson equation for the electrostatic field of a point charge not at the origin,
~
~ ~r .( R̂ ) = 4πδ 3 (R)
∇
R2
(26)
We can use the above procedures given in this section to apply the Gauss’ integral law to any field of the
m
form ( RR̂n ) with n < 3. In this context, we note that an expression of the form ( rrn ) with n ≥ m is not
well-defined at r = 0 although the limit r → 0 may exist. A different derivation is given in [8], where we
have to change the integrand in Eq.(21).
We will now derive an exact expression for the electrostatic energy. The expression for the electrostatic
energy obtained in this section agrees with the standard expressions [7,8], when the sources are not point
charges. However, we will show that, in the potential formulation, the electrostatic self-energy of a point
charge is vanishing instead of being infinite. This is in contrast to the standard expressions [7,8], where the
electrostatic self-energy of a point charge is infinite.
When we start from the potential formulation, the expression of the electrostatic energy, expressed in
terms of the electrostatic potential, is given by the following expression [7,8]:
ξ=
1
2
Z
ρ(~r′ )V (~r′ )dτ
(27)
9
The integral can be taken to be over the complete volume. Here, V (~r′ ) is the potential at the point ~r′ due
to all the other source elements apart from that at the point ~r′ itself. We denote such quantities by VR (~r′ ).
However, the charge density is given by the divergence of the electric field due to the complete source. We
denote such quantities by the suffix T . Thus the energy is given by the following expression:
ǫ0
ξ=
2
Z
~ E
~ T )VR dτ
(∇.
(28)
~ T respectively. The
For a non-singular source density, both VR and E~R can be replaced by VT and E
corresponding fields and the potentials are non-singular, and are vanishing in the limit the source volume
element tends to zero. We can replace VR by VT in Eq.(28) and we can also apply the Gauss’integral law to
~ T ). We then have the conventional expression for the electrostatic energy expressed in
the vector field (VT E
terms of the electric field:
ǫ0
ξ=
2
Z
~ T .E~T )dτ
(E
(29)
~ T is the complete field at the point (x, y, z). However, the situation is different when we have point
Here E
charges. In these cases, VR is finite at the point charges. On the other hand VT is divergent at the point
charges and the value of the energy becomes different if we replace VR by VT in Eq.(28). In fact, replacing VR
by VT in Eq.(28) for point charges, gives us the usual divergent value for the electrostatic self energy of point
charges provided we can apply the Gauss’ integral law to the fields divergent as r13 . We have mentioned at
the beginning of this section, that the application of the Gauss’ law to fields divergent as r13 is a non-trivial
issue.
In this section we evaluate the electrostatic energy from Eq.(28) without replacing VR by VT . When we
have an isolated point charge, VR is zero at the point charge but is finite for any other point. Let us consider
~ T ) varies as 13 but is zero at the point charge. As we
the point charge to be at the origin. The product (VR E
r
have discussed earlier, we can not apply the divergence theorem to such an ill-behaved field. We note that VR
is zero where the delta function representing the source density is non-vanishing. Hence, in Eq.(28), we can
delete an infinitesimal volume element surrounding the point charge from the region of volume integration.
In the limit of zero volume this infinitesimal volume element surrounding the point charge will have zero
contribution. The electrostatic energy is given by the following expression:
ξ′ =
1
2
Z
(V −v)
~ E~T )VR dτ
(∇.
(30)
Here, v is an infinitesimal volume centred at the point charge. We can now apply the divergence theorem
~ T ). The integrand in the above expression, within the annular region of volume
to the vector field (VR E
~ T ). It is easy to find that ξ ′ is vanishing. This is consistent with the value
integration, is given by (VT E
of the energy obtained directly from Eq.(28). We need not to consider any limiting internal structure of
the point charges to explain the self-energy of the point charges. This is considered in [7] to explain the
divergent expression for the electrostatic self-energy. We will discuss this issue again.
We now derive the expression for the electrostatic energy when we have more than one point charge. We
consider the situation with two point charges. The electrostatic energy in terms of the potential is given by
the following expression:
ξ=
1
2
Z
~ E
~ T )VR dτ
(∇.
(31)
In this case, we no longer have VR = 0 at the point charges. Hence, we can not delete the infinitesimally
small volume elements containing the delta functions (the point charges) from our region of integration. We
find that the integral over the delta functions give us the well-known expression for the energy of interaction.
We can delete two infinitesimal volume elements surrounding the point charges from our region of volume
integration, provided we take into account this interaction energy. The expression for the electrostatic energy
with two point charges is then given by the following expression:
1
ξ= [
2
Z
~ E
~ T )VT dτ ] + q1 q2 1
(∇.
4πǫ0 R12
(V −v1 −v2 )
(32)
10
Here, (v1 and v2 ) are two infinitesimally small volume elements surrounding the point charges q1 and q2
respectively. One can show that the contribution from the volume integration is zero. This follows from the
Gauss’ law. We have to be careful about the boundary terms, if we want to evaluate the integral explicitly.
Thus, we find that the electrostatic self energy of a point charge is vanishing. We may consider a point
charge to be the limiting situation of an infinitesimally small volume element whose total charge is always
finite. One can then derive an expression for the total electrostatic energy from Eq.(29). If we neglect the
internal structure of such a source and the corresponding electrostatic energy, we have an expression for
the electrostatic energy which diverges inversely as the radius of the source tends to zero. This expression
is similar to the conventional divergent expression of the electrostatic self-energy of a point charge [7].
However, we should note that the situation with a sphere whose radius tends to zero is different from a
sphere whose radius is exactly zero. In the later situation, the Eq.(29) is no longer valid. This is related
~ T ) diverges as
with the applicability of the Gauss’ integral law to divergent fields. In this case the field (VT E
inverse cube of the distance from the point charges and we may not apply the Gauss’ integral law as is done
to obtain Eq.(29). This is discussed below Eq.(29). The above discussions are significant when we consider
the Lagrangian description of classical electrodynamics. We find that the applicability of the Gauss’ integral
law to singular fields is a serious concern, when we try to construct the stress-tensor in terms of the fields
with point particle sources. However, we can start from the Lagrangian density in terms of the fields directly
and evaluate the stress-tensor from the Lagrangian density. This will give the familiar singular expression
for the self energy of a point charge.
IV.
CONCLUSION
To conclude, in this article we have considered the issue of construction of a covariant derivative operator
in a quantum theory of gravity. We have used a path-integral formulation with the Palatini action and
a canonical quantization approach to illustrate this issue. We have found that we have to use connection
coefficients more general than the Levi-Civita symbols in a general quantum theory of gravity. This is valid
even if we do not introduce the gauge and matter fields. We have also found that the generalized connections
give additional degrees of freedom. This is necessary in a canonical quantization scheme and is characteristic
of a path integral formulation with the Palatini action. These fields can be described in terms of the torsion
tensor and a symmetric second rank covariant tensor together with its partial derivatives. These fields may
become important to explain certain cosmological problems like inflation. Thus, a general quantum theory
of gravity can be described by metric, a symmetric second rank covariant tensor and the torsion tensor
irrespective of coupling with the gauge and matter fields. The Palatini action can be used to study the
quantum theory and we can introduce the gauge and matter fields to study the corresponding interacting
theory. We can consider a perturbation theory by considering the torsion tensor to be small compared to
the Levi-Civita symbols and the aµν fields in Eq.(12) to be small compared to gµν . The additional fields
associated with generalized connections vanish in the classical spacetime manifolds which are solutions of
the Einstein equations with connections given by the Levi-Civita symbols. It is a non-trivial issue to extend
the Maxwell equations when connections are not given by the metric compatible Levi-Civita symbols. This
is important for the strict current conservation law and potential formulation. We will discuss this issue
later. Lastly, we have given an alternate proof of the Poisson theorem in electrostatics and shown that the
self energy of a point charge is zero.
V.
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11
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