text:
\section{Introduction and Summary}\label{first}
There is a renewed interest in Newton-Cartan (NC) geometry which has
been observed in the last  years. The first significant paper was
\cite{Son:2013rqa} which introduced NC to field theory that analyzes
strongly correlated electrons. It was further shown in
\cite{Christensen:2013lma,Christensen:2013rfa} that NC geometry with
torsion naturally emerges as the background boundary geometry in
holography for $z=2$ Lifshitz geometries, for relevant works, see
\cite{Geracie:2014nka,Jensen:2014aia,Hartong:2014pma,Hartong:2015wxa}
and for review and extensive list of references, see
\cite{Taylor:2015glc}. In fact, NC geometry is non-relativistic
background geometry to which non-relativistic field theories can be
covariantly coupled, see for example
\cite{Jensen:2014aia,Geracie:2015dea,Bergshoeff:2015sic,Festuccia:2016caf,Geracie:2015xfa,Festuccia:2016awg}.
In particular, it was shown  in significant paper
\cite{Festuccia:2016caf} how non-relativistic electrodynamics can
couple to the most general NC geometry with torsion. Further,
non-relativistic scalar fields coupled to NC geometry and background
electromagnetic field were also analyzed there.

Since these results are very interesting non-relativistic theories
in NC background  certainly deserve to be studied in more details
further. In this short note we focus on canonical analysis of
non-relativistic scalar field and non-relativistic electrodynamics
coupled to NC geometry. It turns out that this is rather non-trivial
problem with interesting property that the constraints  explicit
depend on time. In more details, we start with the action for
Schr\"{o}dinger field in the background NC geometry and background
non-relativistic electromagnetic field. Such an action was derived
in \cite{Festuccia:2016caf} with the help of null reduction of
complex scalar field in higher dimensional space-time \footnote{Null
reduction was studied in some earlier papers
\cite{Duval:1984cj,Julia:1994bs}.}. Then in order to find
Hamiltonian form of this action we have to impose important
restriction on the NC space-time in the sense  that it has to have a
notion of foliation by spatial surfaces that are orthogonal to one
form $\tau_\mu$ where $\tau_\mu$ is known as clock form. This form
defines a preferred notion of spatial direction at each point and
also arrow of time in the sense that vector field $t^\mu $ is said
to be future directed if it obeys the condition $\tau_\mu t^\mu>0$.
$\tau_\mu$ defines a pointwise notion of spatial direction with the
help of the vectors $w^\mu$ that obey the condition $\tau_\mu
w^\mu=0$. However this notion can be integrated to a local
codimension one subspace when $\tau_\mu$ obeys Frobenious condition
$\tau \wedge d\tau=0$ where $\tau=\tau_\mu dx^\mu$. Then we define
causal space-times as space-times where this condition holds
everywhere, for more detailed analysis and discussion, see for
example \cite{Geracie:2015dea,Geracie:2015xfa}. For such space-time
we will be able to find Hamiltonian for the Schr\"{o}dinger field in
NC background. However we also find that when we write the complex
scalar field in polar form as $\psi=\sqrt{\rho}e^{iS}$ that the
momentum conjugate to $\rho$ is zero which is first primary
constraint of the theory. Further, in case of the momentum conjugate
to $S$ we find that it is determined by second primary constraint
that explicitly depends on time. This is very interesting situation
that deserves careful treatment. For that reason we  perform
analysis of constraint systems with explicit time dependent primary
constraints in appendix \footnote{Discussion of the constraint
analysis with explicit time dependence can be found in
\cite{Gitman:1990qh} however the analysis presented there is
slightly different from ours.}. Taking into account  explicit time
dependence of the constraints we will be able to derive canonical
equations of motion that reproduce Lagrangian equations of motion
which is a nice consistency check.

As the next step we extend this analysis to the case of non-relativistic electrodynamics in   Newton-Cartan background. Since canonical analysis is based on an existence of Lagrangian we start with the non-relativistic electrodynamics action in NC background that is derived using null dimensional reduction \cite{Festuccia:2016caf}. We again restrict to the case of causal space-time and in the first step we determine set of primary constraints which Poisson commute among themselves. This is different situation than in case of the scalar field  where the primary constraints were the second class constraints. Then the requirement of the preservation of the primary constraints gives set of secondary constraints which together with the primary constraints form set of the second class constraints. As a result we find that gauge field and corresponding conjugate momenta can be eliminated from the theory at least in principle. We also determine Lagrange multipliers corresponding to the primary constraints using the equations of motion for gauge field and we show that the resulting equations of motion coincide with the equations of motion derived by variation of action.

Let us outline main results derived in this paper. We obtain  Hamiltonian form of non-relativistic theories on NC background and we
determine physical degrees of freedom. This is very important result since we show that in case of non-relativistic electrodynamics the only physical degree of freedom is the scalar field and conjugate momenta. We also discuss the problem of the constraint structure in case of theories with explicit time dependent constraints.

The structure of this paper is as follows. In the next section (\ref{second}) we review basic facts about NC geometry and introduce an action for Schr\"{o}dinger field in the NC background and background non-relativistic electromagnetic field through null dimensional reduction. Then we perform Hamiltonian analysis of this theory and determine structure of constraints. In section (\ref{third}) we analyze non-relativistic electrodynamics in NC background. We firstly perform canonical analysis of non-relativistic electrodynamics in flat background and then we extend this analysis to the case of non-relativistic electrodynamics in NC background. Finally in appendix (\ref{Appendix})
we study  constrained systems with explicit time dependence and
discuss their properties.

\section{Hamiltonian Analysis of  Schr\"{o}dinger field in NC background}
\label{second}
\subsection{Summary of  Newton-Cartan Geometry}
We start this analysis with the brief review of Newton-Cartan geometry in $d+1$ dimensions.
Newton-Cartan background in $d+1$ dimensions is given by a set of one forms $(\tau_\mu,e_\mu^{ \ a})$ where $a=1,\dots,d$ and where $\mu,\nu=0,1,\dots,d$. We also have one form $M_\mu$. We define inverse vielbeins $v^\mu$
and $e_\mu^{ \ a}$ through the relations
\begin{equation}
v^\mu e_\mu^{ \ a}=0 \ ,  \quad v^\mu \tau_\mu=-1 \ ,
\quad e^\mu_{ \ a}\tau_\mu=0 \ , \quad e^\mu_{ \ a}e_\mu^{ \ b}=\delta_a^b \ .
\end{equation}
The determinant of the $(d+1)\times (d+1)$ matrix $(\tau_\mu,e_\mu^{ \ a})$ is denoted by $e$. With the help of  vierbeins we can construct degenerative "spatial metric"
\begin{equation}
h_{\mu\nu}=e_\mu^{ \ a}e_\nu^{ \ b}\delta_{ab} \ , \quad
h^{\mu\nu}=e^\mu_{ \ a}e^\nu_{ \ b}\delta^{ab} \ .
\end{equation}
By definitions, one forms $\tau_\mu, e_\mu^{ \ a}$ and $M_\mu$
transform under diffeomorphism as usual but they also transform
under various local transformations: Galilean boosts with
$\lambda_a$ as local parameter, local $SO(d)$ rotations which is
parameterized by $\lambda_{ab}=-\lambda_{ba}$ and $U(1)_\sigma$
gauge transformation that is parameterized by $\sigma$ where we have
\begin{eqnarray}
\delta \tau_\mu&=& 0 \ , \quad \delta e_\mu^{ \ a}=\tau_\mu \lambda^a+\lambda^a_{ \ b}e_\mu^{ \ b} \ , \nonumber \\
\delta v^\mu&=&\lambda^a e^\mu_{ \ a} \ , \quad \delta e^\mu_{ \ a}=\lambda_a^{ \ b}e^\mu_{  \ b} \ , \nonumber \\
\delta M_\mu&=&\lambda_a e_\mu^{ \ a}+\partial_\mu\sigma
 \ .\nonumber \\
 \end{eqnarray}
 The inverse vielbein $e^\mu_{ \ a}$ is invariant under local Galilean transformations.
Note that we have an important relation
\begin{equation}
e_\mu^{ \ a}e^\nu_{ \ a}-\tau_\mu v^\nu=\delta_\mu^\nu
\end{equation}
that implies
\begin{equation}
h_{\mu\nu}h^{\nu\rho}=\delta_\mu^\rho+\tau_\mu v^\rho
\end{equation}
which will be useful below. It is also useful to define  objects
that are invariant under local Galilean transformations $\hv^\mu \ ,
\he^\mu_{ \ a} \ , \hh^{\mu\nu}$ and $\Phi$ defined as
\begin{eqnarray}\label{hv}
\hv^\mu&=&v^\mu-h^{\mu\nu}M_\nu \ , \quad \he^\mu_{ \ a}=e^\mu_{ \ a}-M_\nu e^\nu_{ \ b}\delta^{ba}\tau_\mu \ ,
\nonumber \\
\hh_{\mu\nu}&=&h_{\mu\nu}-M_\mu \tau_\nu-M_\nu \tau_\mu \ ,  \quad
\Phi=-v^\mu M_\mu+\frac{1}{2}h^{\mu\nu}M_\mu M_\nu \ .
\nonumber \\
\end{eqnarray}
It is important to stress that $\hh_{\mu\nu}\neq \he_\mu^{ \ a}\he_\nu^{ \ b}\delta_{ab}$. Instead, using the definition of  $\he_\mu^{ \ a}$ given above, we obtain following relation
\begin{eqnarray}
\he_\mu^{ \ a}\he_\nu^{ \ b}\delta_{ab}=
\hh_{\mu\nu}+2\tau_\mu\tau_\nu\Phi \ . \nonumber \\
\end{eqnarray}
Finally note that  hatted objects
 obey following the relations
\begin{equation}
\hv^\mu \he_\mu^{ \ a}=0 \ , \quad \hv^\mu \tau_\mu=-1 \ , \quad
e^\mu_{ \ a}\he_\mu^{ \ b}=\delta_a^b \ .
\end{equation}
After this review of NC geometry we proceed to the Hamiltonian analysis of Schr\"{o}dinger field.
\subsection{Schr\"{o}dinger Field in NC geometry through Null Dimensional Reduction}
We would like to find Hamiltonian formulation of scalar field on Newton-Cartan background  with fixed electromagnetic background.
The most convenient way how to find such an action is to perform null dimensional reduction, see for example  \cite{Geracie:2015dea,Festuccia:2016caf}.
Let us consider  an action  for complex scalar field in
$d+2$ dimension in the form
\begin{equation}
I=\int d^{d+2}x\sqrt{-\gamma}
\left(-\gamma^{AB}D_A \Psi D_B\Psi^*\right) \ ,
\end{equation}
where $D_A=\partial_A \Psi-iq A_A\Psi$ and where $A_A, A=0,\dots,d+1$ is  background
electromagnetic field.
Let us now consider a background metric which possesses a null
isometry that is generated by coordinates $\partial_u$
\begin{equation}
ds^2=\gamma_{AB}dx^A dx^B=2\tau_\mu dx^\mu(du-M_\nu dx^\nu)+h_{\mu\nu} dx^\mu dx^\nu
\end{equation}
so that
\begin{equation}
\gamma_{\mu u}=\gamma_{u\mu}=\tau_\mu \ , \quad \gamma_{\mu\nu}=
h_{\mu\nu}-\tau_\mu M_\nu -\tau_\nu M_\nu\equiv \hh_{\mu\nu}
\ .
\end{equation}
Then
\begin{equation}
\sqrt{-\gamma}=e \ ,
e=\det \left(\tau_\mu, e_\mu^{ \ a}\right) \ .
\end{equation}
Since the metric $\gamma_{AB}$ is non-singular we can easily find inverse
metric with components
\begin{equation}\label{gammainv}
\gamma^{uu}=2\Phi \ , \quad \gamma^{u\mu}=-\hv^\mu \ , \quad  \gamma^{\mu\nu}=h^{\mu\nu} \ ,
\end{equation}
where $\hv^\mu$ and $\Phi$ are defined in (\ref{hv}).
We further presume that the gauge field has the form $A_A=(A_u,A_\mu)=
(\varphi,\bar{A}_\mu-\varphi M_\mu)$.
Now with the help of this metric we perform dimensional reduction of the action.
To do this we have to presume that all fields do not depend on $u$. We impose following ansatz for the scalar field $\Psi$
\begin{equation}
\Psi=e^{imu}\psi
\end{equation}
and insert it to the action. Then also using (\ref{gammainv}) we obtain
\begin{eqnarray}\label{Schroact}
I&=&\int d^{d+1}xe\left(\hv^\mu (D_\mu\psi)^*i(m-q\varphi)\psi-i(m-q\varphi)\hv^\mu D_\mu\psi
\psi^* \right.
\nonumber \\
&-& \left. 2\Phi (m-q\varphi)^2\psi \psi^*-h^{\mu\nu}D_\mu\psi (D_\nu\psi)^*\right) \ ,
\nonumber \\
\end{eqnarray}
where
\begin{equation}
D_\mu\psi=\partial_\mu\psi-iq A_\mu \psi \ , \quad (D_\mu\psi)^*=
\partial_\mu\psi^*+iq A_\mu\psi^* \ .
\end{equation}
The action (\ref{Schroact}) is the action for Schr\"{o}dinger field of the mass $m$ and charge $q$ in
Newton-Cartan background and in the background electromagnetic field
where the electromagnetic field has components $A_\mu$. Note that $\psi$ couples to $\varphi$ through the combination $m-q\varphi$ and hence
$\varphi$ effectively shifts the mass of the scalar field and hence it is natural to call it as mass potential
\cite{Festuccia:2016caf}.
 Our goal is to find Hamiltonian from the action (\ref{Schroact}).
\subsection{Hamiltonian Analysis}
We would like to work with real variables rather than with complex ones. For that
reason se introduce following parameterization of the scalar field $\psi$ as  $\psi=\sqrt{\rho}e^{iS}$ so that the action has the form
\begin{eqnarray}
I^{sch}
&=&\int d^4x e\left(2(m-q\varphi)\rho \hv^\mu \partial_\mu S-
2(m-q\varphi)\hv^\mu A_\mu\rho-2\Phi(m-q\varphi)\rho-\right.\nonumber \\
&-&\left.\frac{1}{4\rho}h^{\mu\nu}(D_\mu\rho)^*D_\nu\rho+2qh^{\mu\nu}\sqrt{\rho}
A_\mu\partial_\nu S-\rho h^{\mu\nu}\partial_\mu S\partial_\nu S\right) \ ,
\nonumber \\
\end{eqnarray}
where
\begin{equation}
D_\mu\rho=\partial_\mu\rho-2iqA_\mu\rho \ , \quad
(D_\mu\rho)^*=\partial_\mu\rho+2iqA_\mu\rho \ .
\end{equation}
 It is clear that  the previous action is well defined for  general Newton-Cartan background
for arbitrary $\tau_\mu$ apart from the fact that $\tau_\mu$ has to
obey Newton-Cartan compatibility condition. On the other hand in
order to have well defined
 Hamiltonian formulation
 we have to have a notion of  foliation by spatial surfaces that are orthogonal to $\tau_\mu$. This is guaranteed when we impose
hypersurface orthogonality condition $\tau_{[\mu}\partial_\nu
\tau_{\rho]}=0$ on the whole space-time $M$. This condition is known
as Frobenius condition and for more detailed discussion   of
causality in Newton-Cartan background, see
\cite{Geracie:2015dea,Geracie:2015xfa}. Space $M$ that obeys this
condition is called as causal. Since $\tau_\mu$ is nowhere non-zero
we can write it as $\tau_0=e^{-\Phi_L}$ where $\Phi_L$ is known as
Luttinger potential. In what follows we restrict to such space-time.
Since $\tau_i=0$ we obtain following consequences
on the form of the metric $h^{\mu\nu}$ thanks to the condition
\begin{equation}
\tau_\mu h^{\mu\nu}=0 \ .
\end{equation}
Explicitly,  for  $\nu=0$ this equation implies $\tau_0 h^{00}=0$ and hence we have to have $h^{00}=0$ while for $\nu=i$ we have
$\tau_\mu h^{\mu i}=\tau_0 h^{0i}=0$ which again implies that $h^{0i}=0$.
Then the action $I^{sch}$ simplifies considerably
\begin{eqnarray}\label{Isch}
I^{sch}&=&\int d^{d+1}x e\left(2(m-q\varphi)\rho \hv^\mu \partial_\mu S-
2(m-q\varphi)\hv^\mu A_\mu\rho-2\Phi(m-q\varphi)\rho-\right.\nonumber \\
&-& \left.\frac{1}{4\rho}h^{ij}(D_i\rho)^*D_j\rho+2qh^{ij}\sqrt{\rho}
A_i \partial_j S-\rho h^{ij}\partial_i S\partial_j S\right) \ ,
\nonumber \\
\end{eqnarray}
where $\hv^\mu=v^\mu -h^{\mu\nu}M_\nu$ has generally non-zero all its components.

Before we proceed to the Hamiltonian formulation of the theory we derive equations of motion for $\rho$ and $S$ from (\ref{Isch})
\begin{eqnarray}\label{eqIschlag}
& &\partial_\mu[e\hv^\mu (m-q\varphi)\rho]+q\partial_i[e\sqrt{\rho}h^{ij}A_j]
-\partial_i[e\rho h^{ij}\partial_jS ]
=0 \ , \nonumber \\
& &2e(m-q\varphi)\hv^\mu\partial_\mu S-2e(m-q\varphi)\hv^\mu A_\mu
-2e\Phi(m-q\varphi)+\nonumber \\
& &+\frac{e}{4\rho^2}(D_i\rho)^*D_j\rho h^{ij}
+D_i^*\left[\frac{e}{4\rho}h^{ij}D_j\rho\right]+
D_i\left[\frac{e}{4\rho}h^{ij}(D_j\rho)^*\right] \nonumber
\\
& &+
\frac{q}{\sqrt{\rho}}eh^{ij}A_i\partial_j S-eh^{ij}\partial_i S
\partial_j S=0 \ .\nonumber \\
\end{eqnarray}
 Now we are ready to proceed to the Hamiltonian formalism.
From (\ref{Isch})
 we obtain following conjugate momenta
\begin{equation}
p_S=\frac{\partial \mL^{sch}}{\partial(\partial_t S)}=2(m-q\varphi)\rho \hv^0 \ , \quad p_\rho=\frac{\partial \mL^{sch}}{\partial(\partial_t \rho)}= 0 \ .
\end{equation}
From these two equations we see that there are two primary constraints
\begin{eqnarray}
\mG_S\equiv p_S-2e(m-q\varphi)\rho\hv^0 \approx 0 \ , \quad
\mG_\rho\equiv p_\rho\approx 0 \  \nonumber \\
\end{eqnarray}
while the bare Hamiltonian  is equal to
\begin{eqnarray}
H_B&=&\int d^d\bx (p_\rho\partial_t\rho+p_S\partial_t S-\mL)=
\int d^d\bx\mH_B\ , \nonumber \\
\mH_B&=&-2e(m-q\varphi)\rho\hv^i \partial_iS+2e(m-q\varphi)\hv^\mu A_\mu\rho
+2e\Phi(m-q\varphi)\rho+\nonumber \\
&+&\frac{1}{4\rho}eh^{ij}
(D_i\rho)^*D_j\rho-2qeh^{ij}\sqrt{\rho}A_i \partial_j S+
\rho e h^{ij}\partial_i S\partial_j S \ .
\nonumber \\
\end{eqnarray}
We see that generally $\mG_S\approx 0 $ and $\mH_B$  explicit depend on time. This is not usual situation and we discuss theory of constraints systems with explicit time dependence in more details in Appendix (\ref{Appendix}).

As the next step we calculate Poisson bracket between
 $\mG_S$ and $\mG_\rho$ and we obtain
\begin{equation}
\pb{\mG_S(\bx),\mG_\rho(\by)}=-2e(m-q\varphi)\hv^0\delta(\bx-\by)\equiv
\triangle_{S\rho}(\bx,\by)
\end{equation}
which show that they are two second class constraints. Note that the inverse matrix has the form  $\triangle^{\rho S}=-\frac{1}{2e(m-q\varphi)\hv^ 0}\delta(\bx-\by)$.
As a result we can eliminate $p_\rho=0$ and $p_S=0$ from the set of canonical variables when we introduce Dirac bracket between $\rho$ and
$S$ defined as
\begin{eqnarray}\label{rhoSdir}
& &\pb{\rho(\bx),S(\by)}_D=\pb{\rho(\bx),S(\by)}
-\int  d^d\bz d^d\bz'\pb{\rho(\bx),\mG_S(\bz)}\triangle^{S\rho}(\bz,\bz')
\pb{p_\rho(\bz'),S(\by)}-\nonumber \\
&-&\int  d^d\bz d^d\bz'\pb{\rho(\bx),p_\rho(\bz)}\triangle^{\rho S}(\bz,\bz')
\pb{\mG_S(\bz'),S(\by)}
=-\frac{1}{2e\hv^0(m-q\varphi)}\delta(\bx-\by) \ . \nonumber \\
\end{eqnarray}
As was explicitly shown in Appendix (\ref{Appendix}),  in the presence of the
time dependent constraints the equations of motion for canonical variables have
the form \footnote{See equation  (\ref{eqtimecon}) in Appendix (\ref{Appendix}).}
\begin{eqnarray}
\partial_t \rho&=&\pb{\rho,H_B}_D-\nonumber \\
&-&\int d^d\bz d^d\bz' \pb{\rho,\mG_S(\bz)}\triangle^{S\rho}(\bz,\bz')\frac{\partial p_\rho}{\partial t}-
\int d^d\bz d^d\bz' \pb{\rho,p_\rho(\bz)}\triangle^{\rho S}(\bz,\bz')\frac{\partial \mG_S(\bz')}{\partial t}=
\nonumber \\
&=&
-\frac{1}{e\hv^ 0(m-q\varphi)}\partial_i[(m-q\varphi )e\hv^i\rho]-
\frac{1}{e\hv^0(m-q\varphi)}\partial_t[e(m-q\varphi)\hv^0]\rho- \nonumber \\
&-&\frac{q}{e\hv^0(m-q\varphi)}\partial_i[e\sqrt{\rho}h^{ij}A_j]
+\frac{1}{e\hv^0(m-q\varphi)}\partial_i[eh^{ij}\rho\partial_j S]  \nonumber \\
\end{eqnarray}
that can be rewritten into more symmetric form
\begin{eqnarray}
\partial_t [ e\hv^0(m-q\varphi)\rho]+\partial_i[e\hv^i(m-q\varphi)\rho]
+\partial_i[e\sqrt{\rho}h^{ij}A_j]-
\partial_i[eh^{ij}\rho\partial_j S] =0
\nonumber \\
\end{eqnarray}
that coincides with the first equation of motion  given in (\ref{eqIschlag}). Let us now proceed to the canonical equation of motion for
$S$
\begin{eqnarray}
& &\partial_t S=\pb{S,H_B}_D-
\int d^d\bz d^d\bz'\pb{S,\mG_S(
    \bz)}\triangle^{S\rho}(\bz,\bz')\frac{\partial \mG_\rho}{\partial t}
\nonumber \\
&=&- \frac{\hv^i}{\hv^0}\partial_i S+\frac{\hv^\mu}{\hv^0} A_\mu+\frac{1}{\hv^0}\Phi-\frac{1}{8\rho^2\hv^0(m-q\varphi)}h^{ij}(D_i\rho)^*D_j\rho
\nonumber \\
&-&\frac{1}{2e\hv^0(m-q\varphi)}D_i\left[\frac{eh^{ij}}{4\rho}D_j\rho\right]
-\frac{1}{2e\hv^0(m-q\varphi)}D^*_i\left[\frac{e h^{ij}}{4\rho}D^*_j\rho\right]
\nonumber \\
&-&\frac{eq}{2\hv^0(m-q\varphi)\sqrt{\rho}}h^{ij}A_i\partial_jS
+\frac{e}{\hv^0(m-q\varphi)}h^{ij}\partial_i S\partial_j S
\nonumber \\
\end{eqnarray}
that can be again rewritten into the form
\begin{eqnarray}
& &2e(m-q\varphi)\hv^\mu\partial_\mu S-2e(m-q\varphi)\hv^\mu A_\mu-
2e(m-q\varphi)\Phi+\nonumber \\
& &+\frac{e}{4\rho^2}h^{ij}(D_i\rho)^*D_j\rho+
D_i\left[\frac{e}{4\rho}h^{ij}D_j^*\rho\right]+
D_i^*\left[\frac{e}{4\rho}h^{ij}D_j\rho \right]
+\nonumber \\
& &+\frac{qe}{\sqrt{\rho}}h^{ij}A_i\partial_j S
-eh^{ij}\partial_i S\partial_j S=0
\nonumber \\
\end{eqnarray}
that coincides with the second equation of motion given in
(\ref{eqIschlag}).

In summary, we found the Hamiltonian formulation of  Schr\"{o}dinger field in NC background.
We found that the dynamical fields are $\rho$ and $S$ that have non-zero Dirac bracket
(\ref{rhoSdir}). Then we derived their canonical equations of motion and found that they coincide with the equations of motion derived from Lagrangian.

\section{Hamiltonian Formalism for Electromagnetic Field in
Newton-Cartan Gravity}\label{third}
In this section we focus on canonical analysis of non-relativistic electromagnetic
field in NC background. We start with the simpler case of the action for non-relativistic electrodynamics in flat background.
\subsection{Non-Relativistic Electrodynamics through Null Dimensional Reduction}
Following   \cite{Festuccia:2016caf}
we derive an action for non-relativistic electrodynamics by
 performing a null reduction of the Maxwell
action in one higher dimension. More precisely, let us consider $d+2$ dimensional
Maxwell action
\begin{equation}
S=-\frac{1}{4}\int dt du d^d\bx F_{AB}\eta^{AC}\eta^{BD}F_{CD}
\ ,
\end{equation}
where $\eta_{AB}dx^Adx^B=2dtdu+dx^idx^i$. Following     \cite{Festuccia:2016caf}
we set $A_u=\varphi \ , A_t=-\tvarphi \ ,  A_i=a_i$ and presume that all fields
do not depend on $u$. Since the inverse metric has the form $\eta^{tu}=\eta^{ut}=1 \ ,
\eta^{ij}=\delta^{ij}$ we get
\begin{equation}
F_{AB}\eta^{AC}\eta^{BD}F_{CD}=-2(F_{tu})^2+F_{ij}F^{ij}-4F_{iu}F_{tk}=
-2(\partial_t\varphi)^2-4(\partial_t a_i+\partial_i\tvarphi)\partial_i\varphi
+f_{ij}F^{ij} \ .
\end{equation}
As a result we obtain an action for non-relativistic electrodynamics in flat
background in the form
\begin{equation}\label{actnonEM}
S=\int dt d^d\bx\left(-\frac{1}{4}f_{ij}f^{ij}+(\partial_i\tvarphi+\partial_t a_i)\partial_i\varphi+\frac{1}{2}(\partial_t \varphi)^2\right) \ ,
\end{equation}
where $f_{ij}=\partial_i a_j-\partial_j a_i$.  From the action (\ref{actnonEM}) we
derive following conjugate momenta
\footnote{In this section we do not carry about upper or lower spatial index since they are equivalent in flat background.}
\begin{equation}
\pi^i=\partial_i \varphi \  , \quad p_{\tvarphi}= 0 \ , \quad
p_\varphi=\partial_t \varphi
\end{equation}
so that we have following primary constraints
\begin{eqnarray}\label{primnonEM}
\mG^i\equiv
\pi^i-\partial_i\varphi \approx 0 \ , \quad  p_{\tvarphi}\approx 0 \ ,
\nonumber \\
\end{eqnarray}
together with the bare Hamiltonian in the form
\begin{equation}
H_B=\int d^d\bx (\pi^i\partial_t a_i
+p_\varphi\partial_t\varphi+p_{\tvarphi}\partial_t \tvarphi-\mL)
=
\int d^d\bx\left(\frac{1}{4}f_{ij}f^{ij}-\partial_i \tvarphi \partial_i\varphi
+\frac{1}{2}p_\varphi^2\right) \
\end{equation}
and consequently the extended Hamiltonian is equal to
\begin{equation}
H_E=H_B+\int d^d\bx(\lambda_i\mG^i+\lambda^{\tvarphi} p_{\tvarphi}) \ ,
\end{equation}
where $\lambda_i,\lambda^{\tvarphi}$ are Lagrange multipliers corresponding to the constraints $\mG^i\approx 0$ and $p_{\tvarphi}\approx 0$.
As the next step we have to ensure the preservation of all primary constraints.
In case of the constraint $\mG^i\approx 0$ we obtain
\begin{equation}
\frac{d\mG^i}{dt}=\pb{\mG^i,H_E}=\partial_k f^{ki}-\partial_i p_\phi \equiv \mG^i_{II}\approx 0 \ ,
\end{equation}
where $\mG^i_{II}\approx 0$ are secondary constraints. In case of the constraint
 $p_{\tvarphi}\approx 0$ we obtain
\begin{equation}
\frac{dp_{\tvarphi}}{dt}=\pb{p_{\tvarphi},H_E}=\partial_i \partial^i\varphi
=\partial_i\pi^i-\partial_i \mG^i\approx \partial_i\pi^i\equiv
\mG_{\tvarphi}^{II}\approx 0 \
\end{equation}
which is the generator of gauge transformations. In fact, if we define
\begin{equation}
\bG(\Lambda)=\int d^d \bx \Lambda \mG^{II}_{\tvarphi}
\end{equation}
we obtain standard transformation rules
\begin{equation}
\pb{\bG(\Lambda),A_i}=\partial_i \Lambda \ , \quad
\pb{\bG(\Lambda),f_{ij}}=0 \ .
\end{equation}
Finally we have to ensure the preservation of the constraint $\mG^i_{II}\approx 0$.
To do this we have to calculate the Poisson bracket between constraints
 $\mG^i$ and $\mG^i_{II}$. After some calculations we obtain
\begin{eqnarray}\label{triangleij}
\pb{\mG^i(\bx),\mG^j_{II}(\by)}=
-\partial_k\partial^k\delta(\bx-\by)\delta^{ij}\equiv
\triangle^{ij}(\bx,\by) \ .
\nonumber \\
\end{eqnarray}
Let us introduce an inverse matrix $D_{ij}(\bx,\by)$
that obeys the relation
\begin{equation}
\int d^d \bz \triangle^{ik}(\bx,\bz)D_{kj}(\bz,\by)=\delta^i_j\delta(\bx-\by) \ .
\end{equation}
Since $\triangle_{ij}$ is given in (\ref{triangleij}) we find that
 $D_{ij}$ is a solution of the
equation
\begin{equation}
\frac{\partial}{\partial x^k}\frac{\partial}{\partial x^k}D_{ij}(\bx,\by)=-
\delta_{ij}\delta(\bx-\by) \ .
\end{equation}
As the next step we determine  canonical equations of motion for $\varphi$ and $p_\varphi$
\begin{eqnarray}\label{eqvarphi}
\partial_t\varphi&=&\pb{\tvarphi,H_E}=
\pb{\tvarphi,H_B}+\pb{\varphi,\int d^d\bz \lambda^i\mG_i(\bz)}=p_\varphi
\nonumber \\
\partial_t p_{\varphi}&=&\pb{p_{\varphi},H_E}=
\pb{p_{\varphi},H_B}+\pb{p_{\tvarphi},\int d^d\bz \lambda^i\mG_i(\bz)}=
-\partial_k\partial_k \tvarphi-\partial_i \lambda^i \ .
\nonumber \\
\end{eqnarray}
Finally the equation of motion for $a_i$ has the form
\begin{equation}
\partial_t a_i=\pb{a_i,H_E}=\int d^d\bz \lambda_j(\bz)\pb{a_i,\mG^j(\bz)}=
\lambda_i
\end{equation}
so that the equation of motion for $p_\varphi$ can be written as
\begin{equation}
\partial_t p_{\varphi}=-\partial_k(\partial_k\tvarphi+\partial_t a_k)=\partial_k
\tilde{E}_k \ ,
\end{equation}
where $\tilde{E}_k=-\partial_t a_k-\partial_k\tvarphi$. If we perform  partial
time derivation  of the first  equation in (\ref{eqvarphi})  and use the second one we obtain
\begin{equation}
\partial_t^2 \varphi=\partial_t p_{\varphi}=\partial_k \tilde{E}_k
\end{equation}
with agreement with the equation (2.11) in \cite{Festuccia:2016caf}.  Further,
if we apply  the partial derivative $\partial_i$ on the first equation in
(\ref{eqvarphi}) we obtain
\begin{equation}
\partial_i\partial_t \varphi=\partial_ip_{\varphi}=\partial_k f^{ki}
\end{equation}
with agree with the second equation in (2.3) in \cite{Festuccia:2016caf}. In the same way we find that the divergence of $\mG^i=0$ implies
\begin{equation}
\partial_i \mG^i=
\mG^{II}_{\tvarphi}-\partial_i\partial^i\varphi \approx
- \partial_i\partial^i\tvarphi=0
\end{equation}
that agrees with the first equation in (2.3)
\cite{Festuccia:2016caf}.
Finally we should determine the Lagrange multiplier $\lambda_i$ using the requirement of the preservation of the constraint $\mG^i_{II}$ but this is not necessary since we know that $\lambda_i=\partial_t a_i$. On the other hand  since $\mG^i\approx 0$ and $\mG^i_{II}\approx 0$ are two second class constraints they can be explicitly solved for $\pi^i$ and $a_i$. In other words there is only one dynamical variable which is $\varphi$ and its conjugate momentum $p_{\varphi}$.

\subsection{Null Reduction of Maxwellian Electromagnetism in NC Background}
We determine action for electromagnetic field in Newton-Cartan background
again with the help of null dimensional reduction, following
\cite{Festuccia:2016caf}. We start with the action
for electromagnetic field in $d+2$ dimensions that has the form
\begin{equation}
S=-\frac{1}{4}\int d^{d+2}x\sqrt{-\gamma}F_{AB}\gamma^{AC}\gamma^{BD}F_{CD} \ .
\end{equation}
Our goal is to dimensionally reduce this action along null isometry so that we will presume that $A_M$  do not depend on $u$. We further write $A_M=(A_u,A_\mu)$ and define $A_\mu\equiv \varphi$.
Since the gauge field transforms under $U(1)$ transformations as
\begin{equation}
A'_A=A_A+\partial_A \Lambda
\end{equation}
it is clear that $\varphi$ is invariant under gauge transformation since $\Lambda$ does not depend on $u$. On the other hand the gauge field $A_\mu$ transform as
\begin{equation}
A_\mu'=A_\mu+\partial_\mu \Lambda \ .
\end{equation}
In order to perform null dimensional reductions we use the components of  metric inverse given in (\ref{gammainv}) and we obtain the action in the form
\begin{equation}\label{SelectroNCi}
S=\int d^{d+1}x
e \left(-\frac{1}{4}F_{\mu\nu}h^{\mu\rho}h^{\nu\sigma}F_{\rho\sigma}-
\Phi \partial_\mu \varphi h^{\mu\nu}\partial_\nu\varphi+
\frac{1}{2}(\hv^\mu\partial_\mu\varphi)^2-\hv^\nu F_{\nu\mu}h^{\mu\sigma}
\partial_\sigma\varphi\right) \ ,
\end{equation}
where
\begin{equation}
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu \ .
\end{equation}
It is convenient to use slightly different form of the action which
depends on $v^\mu$ instead of $\hv^\mu$. Following
\cite{Festuccia:2016caf}
we introduce vector field $\bar{A}_\mu$ defined as
\begin{equation}
A_\mu=\bar{A}_\mu-\varphi M_\mu \ .
\end{equation}
Performing this substitution in the action (\ref{SelectroNCi}) we find
\begin{equation}\label{SelectroNC}
S=\int d^{d+1}x e\left(-\frac{1}{4}h^{\mu\rho}h^{\nu\sigma}\bar{F}_{\mu\nu}
\bar{F}_{\rho\sigma}-h^{\mu\nu}v^\rho
\bar{F}_{\rho\nu}\partial_\mu\varphi+\frac{1}{2}(v^\mu \partial_\mu\varphi)^2\right)
\ ,
\end{equation}
where
\begin{equation}
\bar{F}_{\mu\nu}=\partial_\mu \bar{A}_\nu-\partial_\nu\bar{A}_\nu-
\varphi(\partial_\mu M_\nu-\partial_\nu M_\mu) \ .
\end{equation}
The  action (\ref{SelectroNC})
 will be the starting point for the Hamiltonian formulation of the theory. As we argued above we restrict to causal space-time with non-zero $\tau_0$ only. Then $v^\mu$ has generally non-zero all components with $v^0=-\tau_0$.
In case of causal space-time the action has the form
\begin{equation}\label{actNCelcaus}
S=\int d^{d+1}xe\left(-\frac{1}{4}\bar{F}_{ij}h^{ik}h^{jl}\bar{F}_{kl}-
\Phi \partial_i\varphi h^{ij}\partial_j\varphi+\frac{1}{2}
(v^\mu\partial_\mu\varphi)^2-v^\nu \bar{F}_{\nu i}h^{ij}\partial_j\varphi\right) \ .
\end{equation}
Note that from this action we also obtain equations of motion in the form
\begin{eqnarray}\label{eqelectroNC}
& &2\partial_i[e\Phi h^{ij}\partial_j\varphi]-\partial_\mu[ev^\mu v^\nu\partial_\nu\varphi]+
\partial_j[ev^\nu \bar{F}_{\nu i}h^{ij}]+\nonumber \\
&+&
\frac{1}{2}e(\partial_i M_j-\partial_j M_i)h^{ik}h^{jl}\bar{F}_{kl}
+ev^0(\partial_0 M_i-\partial_i M_0)h^{ij}\partial_j\varphi+
ev^k(\partial_k M_i-\partial_i M_k)h^{ij}\partial_j\varphi
=0 \ , \nonumber \\
& &\partial_j[eh^{il}\bar{F}_{lk}h^{kj}]+\partial_0[ev^0 h^{ij}\partial_j\varphi]
+\partial_k[e v^k h^{ij}\partial_j\varphi]-\partial_k[e v^i h^{kj}\partial_j\varphi]=0 \ , \nonumber \\
& & \partial_i[ev^0 h^{ij}\partial_j\varphi]=0 \ . \nonumber \\
\end{eqnarray}
Let us now proceed to the canonical analysis.
From (\ref{actNCelcaus}) we obtain following
 conjugate momenta
\begin{eqnarray}\label{momenta}
\pi^i&=&\frac{\partial \mL}{\partial (\partial_t \bar{A}_i)}=-ev^0 h^{ij}\partial_j\varphi
 \ , \quad  \pi^0=\frac{\partial \mL}{\partial (\partial_t \bar{A}_0)}\approx 0 \ , \nonumber \\
p_\varphi&=&\frac{\partial \mL}{\partial (\partial_t \varphi)}=ev^0(v^\mu\partial_\mu\varphi)
\nonumber \\
\end{eqnarray}
so that we have following explicitly time dependent primary constraints
\begin{equation}
\mG^i\equiv \pi^i+ev^0 h^{ij}\partial_j\varphi \approx 0
\end{equation}
together with familiar constraint $\mG^0\equiv\pi^0\approx 0$. Further, with the help of
 (\ref{momenta}), we obtain  the bare Hamiltonian in the form
\begin{eqnarray}
H_B
&=&\int d^d\bx \left(e\frac{1}{4}\bar{F}_{ij}h^{ik}h^{jl}\bar{F}_{kl}+e\Phi \partial_i\varphi
h^{ij}\partial_j\varphi+ev^k \bar{F}_{ki}h^{ij}\partial_j\varphi+\right.\nonumber \\
&+& \left.\frac{1}{2e}(\tau_0)^2
p_\varphi^2+\tau_0 v^i\partial_i\varphi
 p_\varphi-A_0\partial_i\pi^i+\pi^i
 \varphi(\partial_0 M_i-\partial_i M_0)\right) \ .  \nonumber \\
\end{eqnarray}
Now we have to analyze the requirement of the preservation of primary constraints
$\mG^i\approx 0 \ , \mG^0\approx 0$. Note that the extended Hamiltonian has the form
\begin{equation}
H_E=H_B+\int d^d\bx (\lambda_i\mG^i+\lambda_0\pi^0) \ .
\end{equation}
 In case of $\mG^0\approx 0$ we obtain that the requirement of its preservation during the time development of the system implies standard
Gauss law constraint
\begin{equation}
\mG^{II}\equiv \partial_i\pi^i\approx 0 \ ,
\end{equation}
while in case of $\mG^i$ we get
\begin{eqnarray}
\frac{d\mG^i}{dt}&=&\frac{\partial \mG^i}{\partial t}
+\pb{\mG^i,H_E}\nonumber \\
&=&\partial_t (ev^0 h^{ij})\partial_j\varphi
-\partial_k [e h^{ik}\bar{F}_{kl}h^{lj}]+\partial_k(e v^k h^{in}\partial_n\varphi)-
\nonumber \\
&-&\partial_m (ev^i h^{mn}\partial_n \varphi)+ev^0 h^{ij}\partial_j [\frac{1}{e} (\tau_0)^2p_{\varphi}]+
ev^0 h^{ij}\partial_j [\tau_0 v^m\partial_m\varphi] \equiv \mG_{II}^i\approx 0 \ ,
\nonumber \\
\end{eqnarray}
where we used the fact that $\pb{\mG^i(\bx),\mG^j(\by)}=0$. We see that the requirement of the preservation of the constraints $\mG^i\approx 0$ implies the second set of the constraints $\mG^i_{II}\approx 0$.
It is again easy to see that $\mG^i\approx 0 \ , \mG^j_{II}\approx 0$ are two sets of second class constraints with rather complicated Poisson bracket between them. Then it is difficult to determine Lagrange multipliers $\lambda_i$
from the requirement of the preservation of the constraints $\mG^i_{II}\approx 0$ during the time evolution of the system. On the other hand, as we will show below, these Lagrange multipliers can be determined with the help of the equations of motion for $\bar{A}_i$.  Further, it is easy to see that $\mG^0$ and $\mG^{II}\approx 0$ are first class constraints where $\mG^{II}\approx 0$ is generator of gauge transformations.

As we argued above $\mG^i\approx 0 \ , \mG^i_{II}\approx 0$ are two sets of second
class constraints where $\mG^i=0$ can be solved for $\pi^i$ while  $\mG^i_{II}=0$ can be solved for $\bar{A}_i$ at least in principle. On the other hand when we try to write equations of motion for $\varphi$ and $p_{\varphi}$ it is convenient to express Lagrange multiplier $\lambda_i$ as a function of  non-dynamical variable $\bar{A}_i$ using its equation of motion
\begin{equation}
\partial_t \bar{A}_i=\pb{\bar{A}_i,H_E}=
\partial_i A_0+\lambda_i+\varphi (\partial_0 M_i-\partial_i M_0)
\end{equation}
that implies that $\lambda_i=\bar{F}_{0i}$.
Then we can  write canonical equations of motion for
$\varphi$ and $p_{\varphi}$ as
 \begin{eqnarray}
 \partial_t \varphi&=&\pb{\varphi,H_B}+\int d^d\bz \lambda_i\pb{\varphi,\mG^i(\bz)}=
 \frac{1}{e(v^0)^2}p_{\varphi}+\tau_0 v^i\partial_i\varphi \ , \nonumber \\
 \partial_t p_{\varphi}&=&\pb{p_{\varphi},H_B}
 +\int d^d\bz \lambda_i(\bz)\pb{p_{\varphi},\mG^i(\bz)}
 =\nonumber \\
 &=&\frac{1}{2}e
 (\partial_i M_j-\partial_jM_i)h^{ik}h^{jl}\bar{F}_{kl}
 -\pi^i(\partial_0 M_i-\partial_i M_0)+\partial_m (\lambda_j h^{jm}e)+\nonumber \\
&+&2\partial_i[e\Phi h^{ij}\partial_j\varphi]+
 \partial_j[ev^k\bar{F}_{ki}h^{ij}]+\partial_i[\tau_0 v^ip_{\varphi}]
 +ev^k (\partial_k M_i-\partial_i M_k)h^{ij}\partial_j\varphi
 \nonumber \\
 &\approx&
 \frac{1}{2}e
 (\partial_i M_j-\partial_jM_i)h^{ik}h^{jl}\bar{F}_{kl}
 +v^0eh^{ij}\partial_j\varphi (\partial_0 M_i-\partial_i M_0)+
 \partial_m (\bar{F}_{0j}h^{jm}ev^0)+
 \nonumber \\
&+&2\partial_i[e\Phi h^{ij}\partial_j\varphi]+
 \partial_j[ev^k\bar{F}_{ki}h^{ij}]+\partial_i[\tau_0 v^ip_{\varphi}]
 +ev^k (\partial_k M_i-\partial_i M_k)h^{ij}\partial_j\varphi \ .
 \nonumber \\
 \end{eqnarray}
 If we combine these two equations together we obtain
 \begin{eqnarray}
& & \partial_t (e(v^0)^2\partial_t\varphi)=\partial_t p_{\varphi}-\partial_t(ev^0v^i\partial_i\varphi)\nonumber \\
& & =\frac{1}{2}e
 (\partial_i M_j-\partial_jM_i)h^{ik}h^{jl}\bar{F}_{kl}
 +ev^0h^{ij}\partial_j\varphi(\partial_0 M_i-\partial_i M_0)+\partial_m (\bar{F}_{0j}h^{jm}ev^0)
 \nonumber \\
& & +2\partial_i[e\Phi h^{ij}\partial_j\varphi]+
 \partial_j[ev^k\bar{F}_{ki}h^{ij}]-\partial_i[e  v^i v^0\partial_t\varphi]
 +ev^k (\partial_k M_i-\partial_i M_k)h^{ij}\partial_j\varphi \
 \nonumber \\
 \end{eqnarray}
 that coincides with the first equation in (\ref{eqelectroNC}). Further, it is easy to see that the second equation in (\ref{eqelectroNC}) coincides with the secondary constraint $\mG^i_{II}=0$. Finally, the last equation in (\ref{eqelectroNC}) is equivalent to the combination of the primary constraint $\mG^{II}$ and $\mG^i$ since
\begin{equation}
\mG^{II}=\partial_i [\mG^i-ev^0 h^{ij}\partial_j\varphi]=
-\partial_i[ev^0 h^{ij}\partial_j\varphi]=0 \ .
\end{equation}
In summary, we have shown that canonical equations of motion and constraints reproduce Lagrangian equations of motion. We have also determined physical degrees of freedom of non-relativistic electrodynamics and we have shown that there are only two phase space  physical degrees left corresponding to $\varphi$ and $p_\varphi$.

 {\bf Acknowledgements:}

I would like to thank M. Chaichian for his comments and remarks. This  work  was
    supported by the Grant Agency of the Czech Republic under the grant
    P201/12/G028.

redpajama_set_name: RedPajamaArXiv


In original box - complete with all 29 pieces & Box. Has been used so there is very minor wear. Has been kept in lovely condition. The item "Steckschlüssel Sortiment 1/4 Inch Drive Hazet 854 Socket Set 29 Pieces In Box" is in sale since Thursday, December 20, 2018. This item is in the category "Business & Industrie\Werkzeuge & Werkstattbedarf\Handwerkzeuge\Sonstige".
The seller is "sevenupone" and is located in WW. This item can be shipped worldwide.


This was the view from the pass that signalled we had almost completed our traverse of the Ganesh Himal on the Great Himalayan Trail on our Pretty Big Walk. Marian is making a shape with the Langtang Himal making a fine fine background.
During the first ten days of walking, we have covered 100 km, made approximately 190,000 steps, and gained about 8700 metres (almost the height of Everest). We've been crossing ridges and valleys and enjoying pretty views almost every step of the way.
Except when it was raining, snowing, hailing, or misting (which it's done most days). Marian and I didn't sleep much on Pangsang Pass (pictured above) because of a ferocious electrical storm that dumped nearly a foot of snow.
We've had sun the past two days with only small pecks of rain. We are hoping the weather pattern has broken and will be more steady for our traverse of the Langtang Himal. Battery power for electronics has been scant so I will close for now.
Thanks for following along and we can't wait to tell you more in a few weeks.
C10. As usual, rain started the moment we hit camp. Water temple nearby for both Hindus & Buddhists. Temple bell rings. Prayer flags release prayers to wind.
L10 After tough climb to pass, rewarded with momentous view of Langtang Himal where we are headed next. Morning sun, nice to dry out after frosty morn.
C7 Misty wisps of cloud surround fern decorated rhododendrons. Roads r bringing rapid change here. Sounds of cows & chainsaws, kharkas & kitchens Did U walk?
L6 view out over three valleys coming together. Evidence of earth folds being pushed up into mtns. Ganesh Mtn reveals its glory. More down now up, up, up!!!
View L4 long down winding stone stair w/ water buffalo. Micro hydro under const. Cicadas sing. Do their ft get sore like ours?
Camp 3 just sky of goal. Jack hammer mixes with shepherds watching goats. Huge day of up, down & distance. Feet yell. Legs cramp. Spirit soars. GHT again.
View from lunch, terraced fields of wheat & corn. Kids curious while adults kind in helping us find our way. Goats passing by. Big up this morn. 1200 m.
Hi from Machakhola. Great hot walk on new road. Life is changing here fast w/ many new roads. Looking across 2 tomorrow's big up. Today broke us in slowly.
Hello from Soti Khola. Setting up camp. The drive had some super muddy parts. The driver was excellent. Ready for a Pretty Big Walk. Day One is in the books.
After a quick day of catching up with friends and errands, the expedition bags are packed and we head off in the morning. We tested our expedition comms this morning and all seem to be working well. As long as battery power holds out, we will send an audio message and location link daily. We likely won't be able to send out many photos so you'll have to depend on my verbal descriptions and use your imagination.
We visited a stupa and enjoyed both people and pigeon watching.
We are traversing the Ganesh Himal and Langtang regions and will be out on the trail for 30 days or so. Depending on weather and snow conditions, (so far there has been lots of both this spring here in the mountains) we may need to alter our route so stay tuned.
We spend much of tomorrow driving to the start of our trip at Soti Khola and then start walking (uphill) on Wednesday. So pleased to have you along!
Marian and I are taking a walk. A Pretty Big Walk. We want you to come along in any way you can…walking, rolling, slowly, quickly, running, jumping, singing…any which way that, gets you, or keeps you moving. We'll be walking, hiking, trekking, and climbing our way across approximately 300 kilometres of the High Himalaya in Nepal.
We'll start our Pretty Big Walk in Chautara on April 10 and head for Tilman Pass Camp and the Langtang Valley using the Great Himalaya Trail (GHT). The Great Himalaya Trail is a series of trails, routes, and primitive paths that allow adventurous folks to walk/climb/trek the entire length of the Himalayas in Nepal.
We'll finish our walk at the beginning of the Manaslu Trek-thus connecting the two walks we've done on the GHT (Great Big Walk in 2014 and Manaslu in 2017). When we finish in mid May, this will mean we have completed five of seven sections of the GHT crossing Nepal You're invited to follow along.


What Are the Benefits of Unsecured Loans for Your Small Business?
Opportunities to work from home have increased exponentially since the expansion of the Internet. Anyone who wants to can work from home nowadays. However, for those who do, creating a sustainable home business environment is essential.
Being a self-employed freelancer in today's working environment carries with it plenty of additional responsibilities that a traditional working environment does not. However, strict rules for managing the working day generally don't work well for these types of business arrangements. Some companies that have implemented a work-from-home policy have had to learn this lesson the hard way.
On the other hand, some studies indicate that working from home leads to increased productivity. However, this kind of working arrangement requires more self-discipline. Here we have gathered some tips to help stay-at-home workers achieve high levels of productivity.
Being your own manager is not easy. You must supervise your own tasks, finances, projects, pitches, goals, and results.
However, structuring your business management with interactive calendars, online reminders, or reporting tools can help you to efficiently handle your workflow. These will help to ensure that your tasks are completed on time. Use them to keep your priorities on track.
Working on a couch might be tempting, but leisure rooms should stay just that—for leisure. Maintaining a separate office space will shift your perception of staying at home. You'll have the feeling of actually going to a job. This will naturally increase your productivity level.
Also, your health will benefit. The ergonomics of well-designed office furniture far surpass those of couches designed for relaxing.
It's good to wake up early and get to working in your home business environment soon after. This can increase your overall effectiveness. Prolonging the waking-up process by taking a long breakfast and concentrating on things that are unrelated to your work can cause lethargy and decrease your productivity.
With all of its perks, the home business environment lacks the benefits of companionship and troubleshooting with co-workers. This can be especially difficult for extroverts. If you're someone who likes to surround yourself with people, organize regular live meetings or turn to technology.
Checking in with coworkers and staying connected with office happenings can easily provide a sense of presence. Instant messaging, video chats, and phone calls can remind you of your overall contribution to the larger operation.
Different tasks require different music. For example, video game soundtracks have proven to be quite effective in increasing productivity levels. This is because they're designed to boost concentration and help players complete tasks. As a matter of fact, even some surgeons listen to music while they work.
However, some people are more productive when there is no noise in their environment. For others, studies show that their productivity depends on which kind of music they listen to.
When you had an office job that you had to show up for every day, you probably had a morning routine. For example, maybe you made coffee and ate breakfast before you dressed up and left your home.
However, when you work from home, routines can quickly deteriorate. You might find yourself lingering in bed. Then perhaps you throw on some sweatpants and a t-shirt instead of dressing in business attire.
Try this instead. Pretend you are going out to a job. Get up early and get dressed up before you go to your desk. You'll find that this will change your mindset. You'll start thinking about work-related tasks as you're preparing to work. And you'll avoid distractions that can affect your productivity.
Dressing up signals your mind that rest time is over and the work day is beginning. You don't have to dress to impress. However, some focus on your appearance can set your mind into the working mode.
With WiFi easily available almost everywhere these days, you can transport your home business environment to a public place. Moreover, this can also help you to shift into the right frame of mind for a workday. Libraries, coffee shops, public lounges, and similar environments can stimulate your energy. They can encourage you to be more productive.
Planning out what you will be working on ahead of time can be a significant productivity booster. Schedules change and adapt, of course. But it is important to have an outline of assignments and an agenda to keep your work on track. Planning will also help you to maximize your business resources and manage your time better.
An essential component for success for people who work from home is keeping finances well managed. Therefore, look online for financial reporting examples that you can utilize for your own operations. These can help you to manage your home business finances and remain productive with your work.
If you share your home space with family or roommates, it is important to set boundaries when you get into your working mode. For some people, this could be a sign on the office door. For others, it might mean putting on headphones.
If there are children in the home, signals resembling traffic lights can help. For instance, for children who are old enough to understand, red would mean that you must not be disturbed. Yellow would be a signal to check with you before barging in. And green would mean it's okay to come into your office.
Communicating your needs to people you share your living (and working) space with is of utmost importance. This is especially true if you have an important online meeting or presentation you need to attend to.
Although it might be tempting to get some work done during the evening or on weekends or holidays, it is crucial to separate working hours from leisure hours. Otherwise, the flexibility that working from home provides can lead to long hours of work with little real rest.
On the other hand, keeping a clear schedule will sustain your energy levels. You'll then have more motivation to get tasks done. And your time for rest will be just that—time for rest.
These are just a few tips for creating a sustainable home business environment. Use them to keep your productivity high and make your work time more fulfilling.
Sandra is a dedicated writer, content manager, and online marketer. Passionate about business intelligence, technology, big data, and analytics, she is working in the buzzing startup scene in Berlin, Germany.


Clement Desalle of the Monster Energy Kawasaki Racing Team reinforced his third place in the FIM World MXGP Motocross Championship with his second podium in the three rounds of the series run to date.
The Belgian turned round a disappointing performance in qualification at the Red Sand facility one hundred kilometres north of the Spanish Mediterranean port of Valencia in a confidence-building performance as he raced in the leading group in both point-scoring races. Rider and team found an excellent solution on the set-up for the starts overnight and he was able to start fourth before quickly taking over third in race one, holding it until an intense mid-race challenge from the series points leader cost him one place in the later stages of the race. Despite not starting quite so well in race two he was quickly able to push forward to third place again and held the position with confidence until a late challenge from the series leader again saw him take the chequered flag in fourth position. This was sufficient to claim third overall on the day and strengthen his third place in the series standings.
KRT teammate Julien Lieber also made his best performance to date in his rookie season in the MXGP class, racing to seventh overall. His first moto ride was particularly impressive as the twenty-three year old put in two strong passes late in the race to finish fifth and he again rode with great confidence in race two to advance from an initial fourteenth after an incident with another rider to take the chequered flag ninth. The two strong performance also sees him move into the series top ten.
Maxime Desprey and Alessandro Lupino of Gebben Van Venrooy Kawasaki Racing each scored nine world championship points in Spain and now hold seventeenth and eighteenth placings in the series.


My son celebrated his 4th birthday in advance on Tuesday at his preschool together with six other children. Every year, I have one big project i.e. my son's birthday. Papa is not in favour of throwing a birthday bash for our little boy. Hence, every year, he would celebrate together with his classmates at the preschool. Thereafter, we will have a mini one with our family members. Last year, he had it at the Singapore Zoo "Breakfast with Orang Utan". This year, I planned it at the Jurong Bird Park, "Lunch with the Parrots". Interestingly, my boy adores animals more than cars. When we talked about going to these two places, his eyes instantly light up and hopping around the house like a rabbit.
I made chocolate cupcakes this year with the 'Jake and the Neverland Pirates' edible image that I ordered from Reeney's baking supplies. This is also a special request from my son. And he also got his wish to put on a red headscarf pretending to be the 'Jake', the pirate. This is my first attempt baking cupcakes and I would say, the buttercream icing, is awfully sweet. But the kids love it and savour every bit of it. The receipe I used is from Joy of Baking. Usually, I would reduce the amount of sugar by 20% or more for all my baking. And I did it for the chocolate cake except for the icing (in order to get the right consistency to pipe on the cupcake.) I have a couple of sleepless night searching the correct size of cupcake case and boxes and worrying about the end result. Moreover, I began to train my boy to sleep in his room but every nights, he would wake me up to sleep together with him. But my efforts pay off, he is finally able to sleep alone in his own room without waking up in the middle of night looking for me. Hooray! Okay, I digress a little... Continue the birthday.. Besides the cupcakes, I also bought 7 Mickey Mouse and Minnie stationery set to be given to each and every kids. And I also promised him to twist balloons for everyone. I managed to do a few simple balloons like pirate sword, doggie, giraffe and hummingbird which I learned from internet. Lastly, I enlisted my Mom's help to buy a box of chocolate in the shape of gold coins from Sheng Siong (because I could not find any in all the candy stores!). This is to match the theme; pirates and gold doubloons. My hubby teased that it is too much for me to handle. But I had proved him wrong because I did it!
On that day, children were excited over the colourful cupcakes and each got to choose their preferred choice of image. The funniest thing was they peeled the edible image off thinking it was sticker. They just love the buttercream icing! And my greedy son has two cupcakes at one go! They were so happy they too received a little gift and gold doubloons from the birthday boy. Oh, yes, the balloons! Everyone went crazy over the balloons. One of his classmate, Jake, was eyeing the balloons and politely asked me "Excuse me, can I have the blue dog?" when everybody were too busy eating their cupcakes to notice it. Clever!
It was a wonderful birthday despite it lasted only 30 minutes. Most importantly, my son and his friends had a fun and great day!


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text:
Canadian Baptist Historical Society
Baptists and Public Life in Canada
Baptists and War
Canadian Baptist Women
Retirement Dinner for Dr. Robert S. Wilson
Posted on May 16, 2018 by canadianbaptisthistoricalsociety
By Taylor Murray
On 2 May 2018, hundreds of people filled the Sheldon L. Fountain Commons at Acadia University to celebrate the career of an extraordinary Canadian Baptist: Dr. Robert S. Wilson (affectionately known to all as "Dr. Bob"). He served as the Dean of Arts and Professor of History at the Atlantic Baptist College (now Crandall University) from 1971 to 1991, and as Professor of Church History at Acadia Divinity College from 1991 to 2018. With the conclusion of the Winter 2018 semester, he retired as the Thomas J. Armstrong Memorial Senior Professor of Church History.
The sizeable crowd was a testimony to the successful career of this important Baptist historian. His work has helped Atlantic Baptists (and Canadian Baptists as a whole) discover their identity. He made important contributions both to town and gown – the pulpit and the pew. As Acadia's "resident layperson," he was always invested in the life of the church and in cultivating strong leaders for the Canadian Baptists of Atlantic Canada. Through his career, he served on nearly every board and council in the Convention, including a term as the President in 1998–1999.
In my conversations with him leading up to the celebration, he ticketed the event as a healthy dose of "toast and roast." Each speaker had his or her own story of appreciation for our dear Dr. Bob. Speakers included representatives from Convention, Crandall University, and Acadia Divinity College; as well as reflections from former students and his children. The Rev. Dr. John Bartol, an elder statesman in the Convention, gave the benediction.
Throughout the evening we were treated to a selection of musical numbers, including several from Dr. Bob's own quartet (pictured below). Also, worth noting, three members of the Acadia community paid tribute to Dr. Bob in an excellent re-written version of Simon and Garfunkel's Sound of Silence.
If there is any truth in the adage, "behind every great man is an even greater woman," I think Dr. Bob would contend it is true in his case. It was a blessing to hear from Myrna (picture above), Dr. Bob's wife of 53 (!) years.
On a personal note, Dr. Bob has meant a lot to our family. He taught my mother (who was one of his first students ever, pictured on his right) and he taught me (one of his last students ever, pictured on his left) – plus he taught my father-in-law. When my mother was his student, he had her babysit his children on several occasions. I thought this was fitting, because years later he was tasked with babysitting me as a lost graduate student!
I am fortunate to have had the opportunity to work with him.
Thank you, Dr. Bob.
Taylor Murray is a PhD student in Church History at McMaster Divinity College. Before coming to MDC, he completed an MA in Christian History at Acadia Divinity College, Acadia University. He is a Member at Large with the Canadian Baptist Historical Society.
**The views of this Blog represent those of the author, and not necessarily the CBHS.**
This entry was posted in Blog, Church History, Photo and tagged Acadia Divinity College, Acadia University, ADC, Baptist History, Canadian Baptist, Canadian Baptist Historical Society, Church History, professor, retirement, Robert S. Wilson, Taylor Murray, Wilson. Bookmark the permalink.
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text:
Reddit Ask-Me-Anything Event
(AMA; March 5, 2012)
Responses by Stephen Wolfram to an "Ask Me Anything" on Reddit.
Q: What is the most interesting use of Mathematica and/or Wolfram Alpha you've ever seen? —Skydiver79
SW: There are so many; very hard to pick just one.
An old one for Mathematica: Mike Foale was using it on the Mir space station; there was an accident; the computer it was on got sucked into space; Mike had a backup disk, but needed a password for a different computer; all-time favorite call to customer service ... and finally an in-action solving of equations of motion for a spinning space station.
Of course, for me personally, my favorite Mathematica "uses" are the research for A New Kind of Science, Wolfram|Alpha ... and the building of Mathematica itself.
Q: Are there any uses for WA that are not typically exploited by users? Any underused functions that we should know about? —justfutt
SW: It's a big challenge letting people know everything that's in Wolfram|Alpha. We try to talk about highlights on blog.wolframalpha.com Probably the best place to look for an overview is www.wolframalpha.com/examples
Q: What was the most difficult technical or design problem that you had to solve during the development of Wolfram|Alpha? —xmachina
SW: I'd been used to building Mathematica, which is a very systematic and coherently designed language, with no visible "heuristics". In Wolfram|Alpha, heuristics are central; our goal is to make it just "do what anyone means". It took me a while to really get into designing a tight system that's so much based on heuristics.
There were many technical issues for Wolfram|Alpha that I thought might just make it all impossible: too much data in the world; too slow to compute useful things; impossible to understand natural language; etc.
Fortunately we got through all of these. One thing to mention is that when one's dealing with natural language, ordinary notions of system modularity tend to go away; a small change in something to do with chemistry might affect some interpretation in finance. It's been interesting to build development and QA systems around all that.
Q: What do you think Wolfram|Alpha will be able to do in 10 years that is impossible today? —anonymous
SW: There'll be quantitative changes in the amount of knowledge in the system that I suspect will have a qualitative effect on how it's used. And there'll be all sorts of computations that become possible because (one assumes) there'll be faster computers that we can use. In Wolfram|Alpha Pro we just started handling input not just of small textual queries, but of data and things like images. As more processing power is available, there'll be some exciting new things to do with those.
Another direction is the ability for Wolfram|Alpha to "invent". Right now it mostly uses existing methods, models, algorithms to compute things. What I'm expecting in the future is that Wolfram|Alpha will be to discover new methods, models and algorithms on the fly. We already do quite a bit of this in our own algorithm development, using ideas I developed in A New Kind of Science. The general idea is to define a task or objective, then search the computational universe for a way to achieve it. The results are often surprising and "clever". An example of this kind of thing is tones.wolfram.com
Another thing that will change a lot in 10 years is the way of accessing Wolfram|Alpha. With Siri, for example, we're seeing voice. There'll be all sorts of interesting directions with augmented reality, etc. etc.
Q: P = NP?
A man can dream... —SheaF91
SW: I suspect that it may be undecidable ... i.e. independent of typical axiom systems.
An interesting approach to it is an empirical one based on enumerating simple programs.
See e.g. https://www.wolframscience.com/nksonline/section-12.8 for the beginning of that. Some more work on this has been done by several people at our NKS Summer School.
Q: on behalf of math majors everywhere, thank you. —cookiemonster87
SW: Thank you!
Q: How did WA end up as the go-to for Siri? —stardog101
SW: Our company and I had a long relationship with Apple and Steve Jobs (see e.g. http://blog.stephenwolfram.com/2011/10/steve-jobs-a-few-memories/)
We'd also started working the Siri team before their company was bought by Apple.
Q: What's some of the uses you've seen of Wolfram Alpha that you never expected when developing it? It's a fantastic "search engine" and I'm constantly amazed at the weird stuff you can pull out of it. —mekarawr
SW: Thanks ... but first, it's not a "search engine" :-) It's not searching anything; it's computing from its built-in supply of computational knowledge.
As we've developed Wolfram|Alpha I've been continually surprised at how much more of the world ends up being computable than I expected. For example, I had no idea that there's be something interesting to compute from a Shakespeare play (try typing in "hamlet").
It's been a repeated experience for me that when I build some big "platform", like Mathematica, or Wolfram|Alpha, I only gradually understand just what the platform makes possible. And that's what's been happening with Wolfram|Alpha. The latest big thing has been with Wolfram|Alpha Pro, starting to understand how the basic ideas of Wolfram|Alpha can be applied not just to short queries, but also to uploaded data, etc.
Q: Hi Stephen,
Thanks for doing this AMA. I had a question in regards to intelligence in children as it relates to their education and socialization. Your wikipedia page states that your intelligence made it difficult to teach you as a child. You were no doubt bored. Was there anything you wish your parents had done differently to make that go smoother as a child? What about social skills? Kids who are much smarter than their peers tend to find it hard to relate or just lack interest in social skills. This makes it hard for them to make and find friends and can lead to self esteem issues in some cases. Was that the case for you? Any advice there?
I ask all these questions because my first baby is due next month. I want to be prepared to handle these types of issues should they arise. Thanks!
As an unrelated question, what do you think is the single most important thing for the US to do in order to regain prominence as a first class educator of children? —anexanhume
SW: I think Wikipedia may overstate the difficulty of my education :-)
I went to some very good schools in England, and typically did rather well. However, starting from probably age 8 or so, I ended up learning the things I was really interested in outside of school, from books, etc. (I wish the web had existed; it would have saved an awful lot of bicycle trips to a library).
I guess I never had much trouble with "self esteem" as such. I had a self image of being a "science type". And that made it a little more difficult to realize that I could and should do things like starting companies.
I think it's often challenging in the educational system for people to understand with clarity (a) what they're really good at, and interested in, and (b) what kinds of niches there are in the world. Too often, people get tracked according to what they happen to do well at early on, and never think outside. In that model, I would have done much less interesting things...
Q: would you consider open sourcing obsolete versions of Mathematica? —Err_Eek
SW: We've thought about things like this from time to time, but it's never seemed to make much sense. It seems like the wrong thing for people to be using obsolete software, and it destroys uniform compatibility of programs written in the Mathematica language ("is it for the obsolete Mathematica, or the real one?", etc.)
A slightly different issue making aspects of Mathematica freely available. We've done that recently with our CDF initiative for computable documents (https://www.wolfram.com/cdf), and it seems to be working well.
For nearly 20 years we've thought about making the "pure language" aspects of Mathematica more freely available (in fact, for example, that was what Sergey Brin worked on when he was an intern at our company long ago...) And I think we may finally soon figure out the right way to do this.
It'll probably be related to my goal in the next year or two of making Mathematica definitively the world's easiest to learn language...
Q: Do you believe that the 'Technological Singularity' is coming, and if so, when? How soon do you think we will see AI pass the Turing test? —anonymous
SW: I think the Turing test will creep up on us. There will be more and more "outsourcing" of human activities (remembering things, figuring things out, recognizing things, etc.) to automated systems. And the line between what's human and what's machine will blur.
For example, I wouldn't be surprised if a future Wolfram|Alpha wouldn't be inserted in the loop for peoples' email or texts: if you want to ask someone a simple question, their "AI" might respond for them.
A thing to understand about AI (that took me a long time to realize): there's really no such thing as "raw general intelligence". It's all just computation—that's one of the big things I figured out in A New Kind of Science (e.g. https://www.wolframscience.com/nksonline/section-12.10). (Actually, it was this observation that made me realize Wolfram|Alpha might be possible now, without us first having constructed a general AI.)
The issue is not to get something "intelligent"; it's to get something with human-like intelligence. And that's all about details of human knowledge and the human condition. Long story ....
Here are a few more thoughts: http://blog.stephenwolfram.com/2011/10/imagining-the-future-with-a-new-kind-of-science/
Q: Where do you see Wolfram Alpha in five years?
What made you guys decide to put in so many amazing easter eggs into it?
What's the breakdown of the Alpha team? I'm curious how many programmers you have, vs. mathematicians and other specialties. Not necessarily absolute numbers if those are secret but proportions would be very interesting to me. —malderi
SW: Five years: answered before.
Easter eggs: they've been a lot of fun for some of the people in the team; I'm glad other people like them too.
Team breakdown:
The main teams we have are: content areas (e.g. socioeconomic; geographic; scientific/medical; math; cultural/consumer; "miscellaneous"); frameworks; parsing; data curation; linguistic curation; user experience/design; web development; quality assurance; operations; [who am I forgetting?]
In aggregate the content areas are the biggest group.
We also have an "advanced R&D group", that works on major new directions (e.g. recently many of the features in Wolfram|Alpha Pro).
In terms of peoples' backgrounds: we've got quite a spectrum, including PhDs in lots of different areas. A remarkable number of people working on core parts of Wolfram|Alpha happen to have worked on NKS—which given some of the methods we're using, may not be a coincidence :-)
For the content areas, we have people with specific expertise and background in those areas (and we're also continually calling on outside experts for help). In other areas, I don't have a precise inventory, but my impression is that we're roughly equally split among physics, computer science and math in terms of educational background.
Q: What is your opinion on Khan Academy? How do you see education in math and science evolving in the next 10 years as computers become even more central? —McMonty
SW: It's been a little frustrating to watch over the years how slowly things in math and science education have been evolving. Back when Mathematica first came out nearly 24 years ago, people started doing things with it in education. And a lot of very nice work has been done.
But I can't help but think there's a lot more that can be done.
Given the current curriculum (e.g. in math) we can do much better at letting individual students move forward at their own pace, e.g. using Mathematica and Wolfram|Alpha as computational engines.
But one thing to realize is that most of the current math curriculum was set up a century ago, when the world was very different. And I strongly believe that it's worth rethinking it, given our current tools, and the current uses that math has in the world.
We have an initiative called "Computer-Based Math" (https://www.computerbasedmath.org) that's exploring this.
One thing that's really nice given Mathematica and Wolfram|Alpha is that people can much more immediately do "real-world" math and science, exploring genuine questions, not toy ones.
Another educational "experiment" of ours is the Wolfram Science Summer School (https://www.wolframscience.com/summerschool/2012/) which we've been running for 9 years. The idea there is that people come and do an original research project. (One of my roles there is a piece of "extreme professoring" ... trying to figure out an appropriate project for each person, given their interests and experience.)
I always start the summer school by doing a "live experiment" with Mathematica, and trying to discover something new in a couple of hours. (So far, it's never failed.) I think it's great for people to see that it's possible to discover new things—and that's then reinforced in the projects they do themselves.
I don't think every teacher is going to be able to pull off making discoveries with live experiments, but I think there are ways to get closer to that.
I've developed quite a few opinions about what the future of at least "high achieving education" should be ... mostly centered on the idea of people being helped in doing "their own projects", rather than being fed standardized courses and curricula.
Gosh ... there's a lot more to say about this. E.g. about treating NKS as a "pre computer science" subject; about teaching Mathematica as a language to young kids (small inputs -> exciting outputs); etc. etc.
Q: What are your opinions on Matlab? —pubby8
SW: Needless to say, I'm not a Matlab user, so I'm not a big expert.
Matlab has certainly at times tried to position itself as a competitor to Mathematica (I'm pretty sure they got the term "technical computing" from a talk I gave—even though I never liked the term in first place).
I haven't run into the authors of Matlab for a very long time ... but my impression is that their goals are rather more modest (at least at a conceptual level) than ours.
My goal with Mathematica has been to cover all areas where systematic computation can be done. And to achieve that, we've built a very general system, based on symbolic programming and symbolic expressions. Matlab is very centered specifically on numerical matrices (hence the name).
Over the years, I've actually been surprised at how much can be turned into a numerical matrix—but ultimately it's a narrow slice of the world, and I think that's now becoming clearer and clearer. In the complete web of algorithms in Mathematica, things that can reasonably be represented as numerical matrices are perhaps 5 or 10% of the total.
By the way, even in terms of numerical matrices, Mathematica is no slouch at this point. 20 years ago Mathematica would have been slower than Matlab at crunching some big numerical matrix. But that's no longer the case ... and in a great many areas, we're able to implement much more advanced algorithms, because in Mathematica we can call on other other capabilities (algebraic, geometric, combinatorial, etc.) to get things done.
Another very important issue is one of philosophy. In Mathematica, my goal has been to make a single coherent system in which one can work, and in which everything fits nicely together. It takes a lot of effort to do this (I've personally spent a large swath of my life doing all the necessary design work). But it's tremendously powerful in actually using (and learning) a system. My impression is that Matlab has taken a different approach, having specific packages that are quite separate (and even bought separately) for different areas, and not really worrying about how they fit together.
Another issue of philosophy is automation. My big idea for Mathematica has been to be able to "delegate" to it as much as possible: I want to just tell the system what I want to do, and I want it to be able to figure out how it should be done. So if there are hundreds of different possible algorithms, I want the system to automatically be able to figure out the best one (unless I happen to feel like tweaking it). In our algorithm development, figuring out how to do this kind of automation is a big part ... but in my experience it's crucial in being able to use a system efficiently.
When I look (which I don't often) at Matlab code, I have to admit to being reminded a little of Fortran (which was one of my first programming languages a very long time ago). Mathematica obviously looks very different (not least because it's a symbolic functional programming language), and even after 25 years, still looks completely modern. (Of course, in Mathematica there's now the quite different possibility of typing pure natural language, which gets interpreted through Wolfram|Alpha.)
There's probably lots more to say here.
One thing I might mention is that closely connected to Matlab is Simulink. We have a major initiative in large-scale systems modeling that I talked a little bit about in: http://blog.stephenwolfram.com/2011/03/launching-a-new-era-in-large-scale-systems-modeling/
There are some pretty exciting things in the works here, linking Modelica modeling with Mathematica and with Wolfram|Alpha. I think the landscape for these kinds of things is going to change a lot in the next few years.
One more thing: as a practical matter, we're seeing more and more of Matlab's traditional engineering users not just being interested in Mathematica, but routinely using Wolfram|Alpha. Again ... I think there are interesting things ahead here.
Q: Stephen, first Thank you! Mathematica/ Alpha are both great. I've got a few questions:
What got you into mathematics in the first place?
What is your favorite piece of mathematics? i.e theorem, proof, fact, construction etc.
Whats the possibility of getting some kind of theorem/proof capabilities into Alpha? I would love to be able to type in a theorem, and get several proofs for it!
—DasCondor
SW:
Actually, I was first interested in physics ... and I learned mathematics as support for that.
I'm not sure if it completely counts as mathematics, but I guess it's the possibility of universal computation. I think that's the most important thing that's been discovered in the past century, and perhaps a lot more.
Well, Wolfram|Alpha obviously is effectively proving theorems in many of the computations it does (e.g. are there solutions to such-and-such an equation?) But if you mean displaying the proofs, that's a somewhat different story.
The "Show steps" buttons for things like college-level integrals are an example of Wolfram|Alpha generating "human understandable explanations" of results it computes.
Mathematica has a fairly powerful general equational logic theorem prover built in, and that can be accessed to some extent from Wolfram|Alpha. We've never figured out a good systematic way to represent proofs in Mathematica ... but it's easier in Wolfram|Alpha, and (though it's not unfortunately a high priority) we will eventually try to do that.
Actually, we have a project that we just started to do "proof-oriented" mathematical structure computations in Wolfram|Alpha. Mathematica works by the user giving input, and Mathematica computing an output "answer".
But in Wolfram|Alpha you can type an input like "caffeine" where there's no specific computation to do; rather one just wants a report. The idea is to do the same kind of thing for math. One might enter "let F be a field with .......". Then Wolfram|Alpha will try to compute "interesting things to say" about that mathematical structure.
It might synthesize new theorems (with heuristics for which ones are "interesting") or it might effectively look up in a computable version of the mathematical literature to see what historical theorems might apply.
Q: Stephen, Why doesn't Mathematica have built in tables of materials properties that are easy to interface with in a problem? For instance, steam tables for water that can be evaluated at any temperature, or materials stress properties as a function of temperature, that can be plugged into any problem just as a variable.
I started off as a physics major, now I am a PhD candidate in nuclear engineering and require these engineering properties. Why isn't Mathematica more engineer friendly? (I'm waiting to be proven wrong-- that these in fact, do exist.)
TLDR: Why aren't there properties tables, which are easy to call and browse, for every possible alloy, chemical, and property? —ZeroCool1
SW: Actually, these capabilities definitely exist in Wolfram|Alpha (e.g. type "water 200C 3 atm").
The Wolfram|Alpha function in Mathematica gets access to them. We're gradually trying to make the access even easier, though.
Q: How is "A New Kind of Science" faring in the scientific community as of lately? —bohrwhore
SW: There's a lot to say here.
There's both interesting science to talk about, and there's interesting history, philosophy and sociology of science.
Later this year, it'll be the 10th anniversary of the publication of A New Kind of Science, and I'm hoping to be able to write some serious assessments at that time.
Personally, pretty much what's happened is what I expected would happen (and even said in the Preface to the book). Some good things have happened quickly, others inevitably take a long time. It's all rather classically "paradigm shifty".
I'm pretty happy with what's in the NKS book, and I continue to be pleased at the number of people who are "discovering" the book, and reading it in remarkable detail.
As with any project this "paradigm shifty", there will be people who think (or at least say) that it's all nonsense. I know that some of the people who made a lot of noise when the book came out have subsequently decided it's a lot more sensible than they at first thought. But I certainly can't say about all of them.
I didn't make a big effort to read all the comments and reviews when the book first came out; in fact I made the conscious decision not to engage in "answering critics" right then.
I'm thinking for the 10th anniversary of reading all the stuff that got written, and then contacting some of the more vigorous critics and setting up some appropriate forum (Reddit ??) to interact with them...
let's define f by
(1-21-s ) f(s) = 1-s - 2-s + 3-s - ...
for complex s with positive real part. How can I find all zeros of this function with Mathematica or Wolfram|Alpha?
Best regards —anonymous
SW: Type "1-s + 2-s + 3-s + ..." into Wolfram|Alpha ... wow! I'm impressed that it can figure out that this is the Riemann zeta function...
Typing "zeros of the riemann zeta function" into Wolfram|Alpha gives some interesting mathematical facts ... but maybe we need a juicy Easter egg about this...
My real question is whether the Riemann Hypothesis is actually decidable in standard axiom systems...
Q: I'm fascinated that you worked alongside Feynman and how you both have such opposite (and complementary) approaches to science.
If it isn't too bold a question, how do you feel about the often circulated letter to you from Feynman? In particular, how do you feel his comments have stood the test of time?
Your projects have been great success and are used worldwide and you've successfully run a large organisation for many years. Have you avoided the managing side of things that Feynman warned you about, or have you enjoyed it? —Lave
SW: I wrote a bit about my interactions with Feynman in:
http://www.stephenwolfram.com/publications/recent/feynman/
Richard Feynman and I worked on quite a few projects together. One example was "quantum computers" back around 1980, way before almost anyone thought about quantum computers.
A typical pattern would be that Feynman would do some elaborate hand calculation, and get some result that I didn't really understand. I would do some computer calculation, and get a result that Feynman didn't understand. And then we would have a big "battle of intuitions" about what each of them meant.
About the letter of Feynman's that you link to: I think it's an interesting letter, though I would claim it's more about Feynman than about me. Feynman himself didn't like doing management kinds of things; he believed he was bad at them, though actually I think he was much better at them than he thought.
I guess I have always liked people, and interacting with people ... and for me managing projects and organizations is interesting and satisfying.
Q: I met you not too long ago at a southern California university. I walked up to you and said hi and thanked you for Wolfram. You spoke to me like I was a moron that didn't deserve your time and brushed me off immediately like I was scum.
I wasn't mad, just disappointed. Hope you have a great life, I still use wolfram all the time! —anonymous
SW: I'm sorry to hear that. I always enjoy chatting with people at events ... in fact, that's a big part of why I do the events. I also pride myself on trying to respond nicely to all kinds of questions and comments...
I actually haven't visited any university in southern California for nearly a decade ... so conceivably you have me mixed up with someone else... !
Q: How are your mental math skills? —wabi44
SW: I always used to think they were terrible ... and that was part of the reason I started a very long time ago using computers to do calculations for me.
However, I have a pretty good memory, and by now I finally know almost every element in the 12x12 multiplication table :-) Also, I think the general level of mental math skills in the population is degrading ... so mine are looking better and better relative to the average.
You might have noticed that Wolfram|Alpha now gives some mental math time estimates for simple computations (e.g. "6+7"). I'm embarrassed to say that I don't always get them in the time it claims for adults... :-)
Q: I went to school in Champaign, IL, where Wolfram Research is headquartered. The rumor was that Stephen Wolfram is amazingly, jaw droppingly, smart. People who are used to being the smartest person in the room are apparently awed by him.
I was hoping I could get confirmation of another rumor from school. Supposedly he would keep a small duffel bag of clothes in his office. If he got stressed out or pissed about something, he would leave work, fly up to Chicago, and then look at the departure board to figure out where on the planet he wanted to go. He would then take an impromptu vacation for a couple days to relax. I always thought that was an awesome idea and I wish I had the money and clout to pull it off. —Big_Baby_Jesus
SW: That story is close to correct, though from long ago. I did this particularly in the mid-1980s when I lived in Princeton, NJ, and People Express airline operated from Newark airport. I ended visiting some rather interesting places...
Since then, my life has gotten considerably more structured ... and I haven't been traveling much at all. Recently, though, my children have decided that I should be traveling more ... because they want to come along. So in the next few months, we're going to some interesting places...
Q: How feasible would it be to give Wolfram Alpha Pro the capability to export plots, tables, etc. into a .tex file or code? I have to do a lot of reports in Latex, and performing the same steps in Matlab over and over to get my plots looking sharp is fairly tedious.
PS: Thank you so much for Wolfram Alpha. This year at university, I have been enrolled in a two-part class that essentially boils down to solving complicated integrals with sometimes up to six dimensions. Being able to plug long expressions into Wolfram Alpha instead of the tedium of figuring it out by hand or sweating proper syntax really saved my sanity. —jwhh91
SW: Thanks for your kind words...
In Wolfram|Alpha Pro, you can export graphics as EPS ... and that should make it easy to include them in TeX documents.
Q: Can you comment on Khan Academy? Awesome answer by the way. —Chanz
SW: I really like the concept of Khan Academy, and the whole idea of "self service education". This is clearly the future. I don't think we've yet hired anyone whose complete education came from places like this ... but I'm sure it will happen.
I've enjoyed meeting Sal Khan, and it's wonderful to see his enthusiasm for the subjects he covers. It's also nice to see Wolfram|Alpha making cameo appearances in recent videos, alongside "pen and paper". (Hmmm ... now I'm thinking about how this might relate to the handwriting-based Wolfram|Alpha that was demoed last week by Samsung on their new tablets at Mobile World...)
Q: I really hate the requirement to log in. I find it very inconvenient, and don't want to do it. The end result is that I try to get much of the same information on Google, where I'm not forced to register to perform a search. —Moruitelda
SW: I'm a bit confused by this. You absolutely can use Wolfram|Alpha without logging in or registering in any way.
But if you want even basic personalized features (like history, favorites, etc.) you obviously need to log in.
We're hoping lots of people will want to upgrade to the Pro version (and it seems to be off to a very good start).
I hope other people agree, but I think we're providing rather impressive value for $2.99/month for students.
In a perfect world, should we try to make every feature absolutely free for everyone? Maybe. I've certainly spent a huge amount on the development of Wolfram|Alpha, and on trying to make as widely available as possible. But to be able to accelerate its development, it needs to start to be subsidized by at least its heavy users.
Ultimately it's a fairly simple proposition: the more successful Wolfram|Alpha can be commercially, the more it can be developed, and the more useful it will be to people.
Thanks to everyone for their support!
Contact|© Stephen Wolfram, LLC


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