Covariance with respect to canonical transformations - a unification of
gravity and electromagnetism?
Gyula Bene
Department of Theoretical Physics, Institute of Physics, Roland E??tv??s University
1117 Budapest, P??zm??ny P??ter s??t??ny 1/A
Plan of the talk:
Introduction
Geometry of the phase space
Calculating connections
Search for field equations
What is still missing?
Conclusion
Introduction
General relativity: covariance with respect to arbitrary coordinate
transformations.
Transformation of coordinates and momenta:
\begin{eqnarray}
x^j&=&f^j(x')\\
p'_j&=&\frac{\partial f^k(x')}{\partial x'^j}p_k
\label{e1}\end{eqnarray}
This is a canonical transformation with generating function
\begin{eqnarray}
W(x',p)=p_jf^j(x') \;.
\label{e2}\end{eqnarray}
Classical electrodynamics: covariance with respect to gauge
transformations
\[A'_j=A_j+\frac{\partial \psi(x')}{\partial x'^j}\]
Transformation of coordinates and momenta:
\begin{eqnarray}
x^j&=&x'^j\\
p'_j&=&p_j+Q\frac{\partial \psi(x')}{\partial x'^j}
\label{e3}\end{eqnarray}
This is a canonical
transformation with generating function
\begin{eqnarray}
W(x',p)=p_jx'^j+Q\psi(x') \;.
\label{e4}\end{eqnarray}
Thus, Nature is covariant with respect to canonical transformations
linear in momenta. Can we generalize this to arbitrary canonical
transformations?
Classical mechanics is already covariant with respect to any
canonical transformations. What about fields?
Realizing an old dream? (Faraday (1850), Einstein (1950))
New physics? (cosmology, dark matter, dark energy)
Approach: along the lines of general relativity. Universal equations
of motion in phase space \(\rightarrow\) free motion in a nontrivial
geometry.
Equation of motion in the presence of gravitational and
electromagnetic fields:
\begin{eqnarray}
m\left(\frac{du^j}{ds}+\gamma^j_{\phantom{j}kl}u^ku^l\right)=QF^{jk}u_k
\label{p1}\end{eqnarray}
\(m\) : rest mass, \(\gamma^j_{\phantom{j}kl}\) : Christoffel's
symbol (associated with the metric \(g_{ik}\)),
\(u^j=\frac{dx^j}{ds}\) : four velocity,
\(F_{jk}=\frac{\partial A_k}{\partial x^j}-\frac{\partial A_j}{\partial x^k}\) :
electromagnetic field tensor.
Hamiltonian:
\begin{eqnarray}
H=\sqrt{g^{ik}(p_i-A_i)(p_k-A_k)}
\label{p4}\end{eqnarray}
we have
\begin{eqnarray}
\frac{dx^j}{ds}&=&\frac{\partial H}{\partial p_j}=\frac{g^{jk}(p_k-A_k)}{H}\;,\label{p5a}\\
\frac{dp_j}{ds}&=&-\frac{\partial H}{\partial x^j}=\frac{g^{ik}(p_k+A_k)A_{i,j}-\frac{1}{2}g^{ik}_{,j}(p_i+A_i)(p_k+A_k)}{H}\;.
\label{p5b}\end{eqnarray}
\(H\) is a constant of motion and its value is \(m/Q\).
\(p_j\) : canonical momentum divided by charge \(Q\).
\(p_j\) and \(A_j\) are measured in units \(c/\sqrt{8\pi \epsilon_0 G}\).
Einstein-Maxwell equations (in the absence of charges):
\begin{eqnarray}
r_{jk}-\frac{1}{2} r g_{jk}&=&-F_{jl}F_k^{\phantom{k}l}+\frac{1}{4}F_{lm}F^{lm}g_{jk}\;,\label{p6a}\\
F^{jk}_{;k}&=&0\;.
\label{p6b}\end{eqnarray}
\(r_{jk}\) : Ricci tensor, \(r\) : Ricci scalar.
Geometry of the phase space
Vectors and tensors in phase space
Phase space coordinates:
\[z^0=x^0\;,\] \[z^1=x^1\;,\] \[z^2=x^2\;,\] \[z^3=x^3\;,\] \[z^4=p_0\;,\]
\[z^5=p_1\;,\] \[z^6=p_2\;,\] \[z^7=p_3\;.\]
Canonical transformation:
\begin{eqnarray}x^j=\frac{\partial \Psi}{\partial p_j}\label{e5}\end{eqnarray}
and
\begin{eqnarray}p'_j=\frac{\partial \Psi}{\partial x'^j}\;,\label{e6}\end{eqnarray}
where \(\Psi=\Psi(x',p)\).
Contravariant vectors
transform like \(dz^j\), i.e.,
\begin{eqnarray}dz'^j=M^j_{\phantom{j}k}dz^k\;.\label{e7}\end{eqnarray}
\(M\) may be expressed in terms of the second derivatives
of the generating function \(\Psi(x',p)\). It satisfies
\begin{eqnarray}MJM^T=J\;,\label{e8}\end{eqnarray}
where
\begin{eqnarray}J^{x^jx^k}=J^{p_jp_k}=0\label{e9}\end{eqnarray}
and
\begin{eqnarray}J^{x^jp_k}=-J^{p_kx^j}=\delta^j_k\;.\label{e10}\end{eqnarray}
Hence, the symplectic unit
matrix \(J\) is a second order contravariant tensor that is left unchanged by
canonical transformations.
Covariant vector components transform like the gradient
of a scalar, i.e.
\begin{eqnarray}\frac{\partial \Phi}{\partial z'^j}=\left(M^{-1}\right)^k_{\phantom{k}j}\frac{\partial \Phi}{\partial z^k}\label{e11}\end{eqnarray}
Obviously, the product of a covariant vector with a contravariant one is a
scalar, therefore Hamilton's equations may be written in the manifestly covariant form
\begin{eqnarray}\frac{dz^j}{ds}=J^{jk}\frac{\partial H}{\partial z^k}\;.\label{e12}\end{eqnarray}
A covariant index may be raised by \(J\), namely
\begin{eqnarray}A^j=J^{jk}A_k\;.\label{e_raise}\end{eqnarray}
This definition, when applied to the (yet undefined) covariant counterpart of
\(J\) itself, yields
\begin{eqnarray}J^{jk}J_{km}=J^{j}_{\phantom{j}m}\;.\label{e_Jraise}\end{eqnarray}
Lowering indices is performed by
multiplication with the covariant tensor \(J_{mk}\). The rule is
\begin{eqnarray}A_j=J_{kj}A^k\;.\label{e_lower}\end{eqnarray}
Consistency requires:
\begin{eqnarray}J^{jm}J_{km}=\delta^j_k\label{e_consis}\end{eqnarray}
Hence, the covariant symplectic unit tensor
\(J_{jk}\) as the inverse transposed of \(J\), which is \(J\) itself. This means that the components of
\(J_{jk}\) are the same as those of \(J^{jk}\). We also have
\begin{eqnarray}J^{j}_{\phantom{j}k}=-\delta^j_k\label{e_mixa}\end{eqnarray}
and
\begin{eqnarray}J_{k}^{\phantom{k}j}=\delta^j_k\label{e_mixb}\;.\end{eqnarray}
The rules imply
\begin{eqnarray}A^jB_j=-A_jB^j\label{e_scal1}\;.\end{eqnarray}
Especially, for any vector \(A^j\)
\begin{eqnarray}A^jA_j=0\label{e_scal2}\;.\end{eqnarray}
Covariant derivatives and Christoffel's symbols
Albeit the tensors \(J^{jk}\) and \(J_{jk}\) which raise and lower indices are constant in this case, unlike
the metric tensor in general relativity, the geometry of phase space is still nontrivial,
because the transformation rule depends on the chosen point in phase space.
This implies that the partial derivative of a vector will not be a tensor,
because it stems from the difference of vectors at different phase space
points. Indeed, differentiating the transformation rule of a covariant vector
component \(v_j\)
\begin{eqnarray}v'_j= \frac{\partial z^k}{\partial z'^j}v_k\label{e13}\end{eqnarray}
one gets
\begin{eqnarray}v'_{j,l}\equiv \frac{\partial v'_j}{\partial z'^l}= \frac{\partial z^k}{\partial
z'^j}\frac{\partial z^n}{\partial z'^l}\frac{\partial v_k}{\partial
z^n}+\frac{\partial^2 z^k}{\partial z'^j\partial z'^l}v_k \;.\label{e14}\end{eqnarray}
Covariant derivatives (denoted by semicolon) that are tensors can be obtained by introducing suitable
Christoffel's symbols
\(\Gamma^k_{\phantom{k}jl}\):
\begin{eqnarray}v_{j;l}=v_{j,l}-\Gamma^k_{\phantom{k}jl}v_k\;.\label{e15}\end{eqnarray}
This implies the transformation rule of Christoffel's symbols:
\begin{eqnarray}\Gamma'^k_{\phantom{k}jl}=\frac{\partial z'^k}{\partial z^s}\frac{\partial
z^r}{\partial z'^j}\frac{\partial z^n}{\partial
z'^l}\Gamma^s_{\phantom{s}rn}+\frac{\partial z'^k}{\partial z^s}\frac{\partial^2
z^s}{\partial z'^j\partial z'^l}\label{e16}\end{eqnarray}
Since the symplectic unit matrix is constant, it is natural to assume that its
covariant derivative vanishes:
\begin{eqnarray}J_{ij;l}=-\Gamma^k_{\phantom{k}il}J_{kj}-\Gamma^k_{\phantom{k}jl}J_{ik}=0\label{e19}\end{eqnarray}
This is equivalent with
\begin{eqnarray}\Gamma_{ijl}=\Gamma_{jil}\;.\label{e20}\end{eqnarray}
(288 independent components)
Let
\[W_{ijk}=\Gamma_{ijk}-\Gamma_{ikj}\]
This is a tensor. Further, it satisfies
\[W_{ijk}+W_{jki}+W_{kij}=0\]
(168 independent components)
Let
\[S_{ijk}=\frac{1}{3}\left(\Gamma_{ijk}+\Gamma_{jki}+\Gamma_{kij}\right)\]
the symmetrized part of the connection (120 independent components). This is not a tensor and may be set to
zero at a given phase space point by a suitable canonical transformation.
By adding the equations
\begin{eqnarray}
\Gamma_{ijk}+\Gamma_{jki}+\Gamma_{kij} &=& 3S_{ijk}\\
\Gamma_{ijk}-\Gamma_{ikj} &=& W_{ijk}\\
\Gamma_{jik}-\Gamma_{jki} &=& W_{jik}
\end{eqnarray}
we get
\[\Gamma_{ijk}=S_{ijk}+\frac{1}{3}\left(W_{ijk}+W_{jik}\right)\]
Another kind of covariant derivative can be defined if the tensorial part is
left out:
\[A_{i:j}=A_{i,j}-S^{k\phantom{ij}}_{\phantom{k}ij}A_k\]
Note that \(A^i_{:i}=A^i_{,i}\) and
\(F^{jk}_{\phantom{jk}:k}=F^{jk}_{\phantom{jk},k}\) if \(F^{jk}\) is antisymmetric.
Calculating connections
Hamilton's equations of motion:
\begin{eqnarray}\frac{dz^i}{ds}=v^i\label{g1}\end{eqnarray}
with
\begin{eqnarray}v^i=J^{ij}\frac{\partial H}{\partial z^j}\label{g1a}\;.\end{eqnarray}
We require this to be identical with the geodetic equation
\begin{eqnarray}\frac{d^2z^i}{ds^2}+\Gamma^i_{\phantom{i}jk}\frac{dz^j}{ds}\frac{dz^k}{ds}=0\;.\label{g2}\end{eqnarray}
Applying Hamilton's equations twice we have
\begin{eqnarray}\frac{d^2z^i}{ds^2}=v^i_{,j}v^j\label{g1b}\;.\end{eqnarray}
Comparing this with the geodetic equation we get
\begin{eqnarray}v^i_{,j}v^j+\Gamma^i_{\phantom{i}jk}v^jv^k\equiv
v^i_{;j}v^j=0\label{g3}\;.\end{eqnarray}
This allows the systematic calculation of the connections:
\begin{eqnarray}
\Gamma_{p_ip_jp_k} &=& 0\\
\Gamma_{p_ip_jx^k} &=& -\frac{g^{ij}(p_k-A_k)}{H^2}\\
\Gamma_{p_ix^kp_j} &=& \Gamma_{x^kp_ip_j}=0\\
\Gamma_{x^ix^jp_k} &=& \gamma^{k}_{\phantom{k}ij}\\
\Gamma_{x^ip_kx^j} &=& \Gamma_{p_kx^ix^j} = \gamma^{k}_{\phantom{k}ij}-\frac{(p_j-A_j)(p^k-A^k)_{;i}}{H^2}\\
\Gamma_{x^ix^jx^k} &=&
-\frac{(p_n-A_n)_{;i}(p^n-A^n)_{;j}(p_k-A_k)}{H^2}\\
&&-\frac{1}{6}\sum_{all\;
permutations\;of\;i,j,k}(p_i-A_i)_{;j;k}\\
&&+\frac{1}{3}(A_{k,j}-A_{j,k})_{;i}+\frac{1}{3}(A_{k,i}-A_{i,k})_{;j}
\end{eqnarray}
The tensorial part is now easily obtained:
\begin{eqnarray}
W_{p_ip_jp_k} &=& 0\\
W_{p_ip_jx^k} &=& -W_{p_ix^kp_j} = -\frac{g^{ij}(p_k-A_k)}{H^2}\\
W_{x^kp_ip_j} &=& 0\\
W_{x^ix^jp_k} &=& -W_{x^ip_kx^j} = \frac{(p_j-A_j)(p^k-A^k)_{;i}}{H^2}\\
W_{p_kx^ix^j} &=& \frac{(p_i-A_i)(p^k-A^k)_{;j}}{H^2}-\frac{(p_j-A_j)(p^k-A^k)_{;i}}{H^2}\\
W_{x^ix^jx^k} &=&
(A_{k,j}-A_{j,k})_{;i}\\
&&+\frac{(p_n-A_n)_{;i}(p^n-A^n)_{;k}(p_j-A_j)}{H^2}\\
&&-\frac{(p_n-A_n)_{;i}(p^n-A^n)_{;j}(p_k-A_k)}{H^2}
\end{eqnarray}
Note that it implies
\[W_{ik}^{\phantom{ik}k}=\frac{2H_{,i}}{H}\]
The symmetrized part of the connection:
\begin{eqnarray}
S_{p_ip_jp_k} &=& 0\\
S_{p_ip_jx^k} &=& S_{x^kp_ip_j}=S_{p_ix^kp_j}=-\frac{1}{3}\frac{g^{ij}(p_k-A_k)}{H^2}\\
S_{x^ix^jp_k} &=& S_{x^ip_kx^j} = S_{p_kx^ix^j} = \gamma^{k}_{\phantom{k}ij}\\
&&-\frac{1}{3}\frac{(p_i-A_i)(p^k-A^k)_{;j}}{H^2}-\frac{1}{3}\frac{(p_j-A_j)(p^k-A^k)_{;i}}{H^2}\\
S_{x^ix^jx^k} &=&
-\frac{1}{3}\frac{(p_n-A_n)_{;i}(p^n-A^n)_{;j}(p_k-A_k)}{H^2}\\
&&-\frac{1}{3}\frac{(p_n-A_n)_{;j}(p^n-A^n)_{;k}(p_i-A_i)}{H^2}\\
&&-\frac{1}{3}\frac{(p_n-A_n)_{;k}(p^n-A^n)_{;i}(p_j-A_j)}{H^2}\\
&&-\frac{1}{6}\sum_{all\;
permutations\;of\;i,j,k}(p_i-A_i)_{;j;k}
\end{eqnarray}
Search for field equations
We are looking for a set of equations that determines \(W\), \(S\) and \(H\)
(altogether 289 field components) so that the resulting equations are
compatible with the Einstein-Maxwell system of equations. Strategy: compiling
all possible Lagrangian densities and finding their right combination. Field
equations are the Euler-Lagrange equations
\[\frac{\partial L}{\partial W_{ijk}}-\frac{\partial}{\partial
z^l}\frac{\partial L}{\partial W_{ijk,l}}=0\]
\[\frac{\partial L}{\partial S_{ijk}}-\frac{\partial}{\partial
z^l}\frac{\partial L}{\partial S_{ijk,l}}=0\]
\[\frac{\partial L}{\partial H}=0\]
These turn out to be tensorial equations with the same symmetries as \(W\) and
\(S\). The last equation is a scalar.
Since \(S\) is not a tensor, it cannot directly enter the Lagrangian
density. We define the Riemannian by
\begin{eqnarray}A_{i:j:k}-A_{i:k:j}=A_lR^l_{\phantom{l}ijk}\label{e24}\end{eqnarray}
This implies:
\[R_{ijkl}=S_{ijl,k}-S_{ijk,l}+S_{ikm}S^m_{\phantom{m}jl}-S_{ilm}S^m_{\phantom{m}jk}\]
Symmetries: \(R_{ijkl}=R_{jikl}=-R_{ijlk}\;,\quad R_{ijkl}+R_{iklj}+R_{iljk}=0\;.\)
Building blocks of the Lagrangian density: \[H\;,\quad W_{ijk}\;,\quad W_{ijk:l}\;,\quad W_{ijk:l:m}\;,\quad R_{ijkl}\;,\quad R_{ijkl:m}\;.\]
What is still missing?
Finding field equations (the right combination)
Source terms and their general derivation in phase space.
Measurable prediction of possible new phenomena.
Conclusion
Motion in gravitational and electromagnetic fields may be
interpreted as free motion in curved phase space
Connections describing phase space geometry are explicitly
expressed in terms of usual electromagnetic and gravitational
quantities
Phase space field equations are still to be found
In case of success what does it mean? Possibilities:
A unified way of formulating gravity and electromagnetism (but
no new phenomena)
A unified way of formulating gravity and electromagnetism
that leads to measurable corrections (new phenomena)
New degrees of freedom may mean new physics far beyond
gravity and electromagnetism.
Electromagnetic interaction has already been unified with weak
interaction. If electromagnetic interaction can also be unified with
gravity, how to include weak interaction?