And this indicates that the “on shell” states form orthonormal basis, provided
we make use of the aforesaid induced inner product.

The coherent state representation of $|\Psi;phy)$ (4.101) corresponding to
(4.107) then yields

$\Psi_{phy}(x,t)=\left(x,t|\Psi;phy\right)=\int
dp~{}(x,t|p,E_{p})~{}(p,E_{p}|\Psi)$ (4.106)

We now consider a Gaussian wave function for the free particle in momentum
space as:

$(p,E_{p}|\Psi)~{}\equiv\Psi(p,E_{p})=\frac{\sqrt{\sigma}}{\pi^{1/4}}\,e^{-\frac{\sigma^{2}p^{2}}{2}}$
(4.107)

The corresponding coherent state representation of wave-packet yields

$\Psi(x,t)=\left(x,t|\Psi;phy\right)=\frac{\sigma^{1/2}}{2\pi^{5/4}}\,\int
dp\,e^{-\frac{\theta p^{4}}{16m^{2}}-\lambda p^{2}+ipx},$ (4.108)

where

$\lambda=(\frac{\sigma^{2}}{2}+\frac{\theta}{4}+i\frac{t}{2m}).$ (4.109)

If only first order in $\theta$ deformations ($\mathcal{O}(\theta)$) are
considered, then one can show that $\Psi(x,t)$ can be recast in the following
simple-looking form,

$\Psi(x,t)\simeq\frac{1}{2\pi^{3/4}}\sqrt{\frac{\sigma}{\lambda}}\,\,\left[1+\theta\,f(x;\lambda)\right]e^{-\frac{x^{2}}{4\lambda}}$
(4.110)

where, the function $f(x;\lambda)$ is

$f(x;\lambda)=\frac{1}{16m^{2}}\left(-\frac{3}{4\lambda^{2}}+\frac{3x^{2}}{4\lambda^{3}}-\frac{x^{4}}{16\lambda^{4}}\right)$

exhibiting a deviation in the functional form, away from Gaussian shape in
coordinate space. The $\theta$-deformation then appears both in the
exponential factor as well as in the amplitude. This clearly demonstrates that
the width $d$ of the deformed Gaussian term at a later time $t$ gets enhanced
due to non-commutativity:

$d=\sqrt{2\mid\lambda\mid}=\left[\left(\sigma^{2}+\frac{\theta}{2}\right)^{2}+\left(\frac{t}{m}\right)^{2}\right]^{\frac{1}{4}}$
(4.111)

This further indicates that even for an infinite spread in the Gaussian wave
packet (4.107) in the momentum space ($\sigma\to 0$), and the spread in
coordinate space $x$ cannot be squeezed bellow the length scale
$\sim\sqrt{\frac{\theta}{2}}$.

###  4.5 Schrödinger equation and energy spectra of harmonic oscillator

In this section, we start by considering the operatorial version of the
Schrödinger equation (4.78) for the time-independent harmonic oscillator
potential $V(\hat{X})=\frac{1}{2}m\omega^{2}\hat{X}^{2}$ :

$[\hat{P}_{t}+\hat{H}]|\Psi;phy)_{H.O}=0,$ (4.112)

where $\hat{H}=\frac{\hat{P}^{2}_{x}}{2m}+\frac{1}{2}m\omega^{2}\hat{X}^{2}$
and let $|\Psi;phy)_{H.O}$ be a physical states corresponding to the harmonic
oscillator system. Now the coherent state representation of the above equation
(4.113) yields

$i\frac{\partial}{\partial
t}\Psi_{H.O}(x,t)=[-\frac{1}{2m}\frac{\partial^{2}}{\partial^{2}x}+V(X_{\theta})]\Psi_{H.O}(x,t)$
(4.113)

where we have used the simpler notation $(x,t|\Psi;phy)_{H.O}=\Psi_{H.O}(x,t)$
for brevity.

Introducing a non-unitary operator

$S=e^{\frac{\theta}{4}(\partial^{2}_{t}+\partial^{2}_{x})}e^{-i\frac{\theta}{2}\partial_{t}\partial_{x}},$
(4.114)

and after some algebra, one can easily show that the space-time coordinate
representation $X_{\theta}$ (4.73) and $T_{\theta}$ (4.74) can be casted in
simpler forms as,

$X_{\theta}=x+\frac{\theta}{2}(\partial_{x}-i\partial_{t})=SxS^{-1};$ (4.115)

$T_{\theta}=t+\frac{\theta}{2}(\partial_{t}+i\partial_{x})=S^{\dagger}t(S^{\dagger})^{-1}$
(4.116)

respectively.

Now, for time independent potential $V(X_{\theta})$, it is straightforward to
show that

$V(X_{\theta})=V(SxS^{-1})=SV(x)S^{-1}$ (4.117)

Using the relation (4.115) and (4.117) we can show the above Schödinger
equation (4.113) can be rewritten as

$i\partial_{t}\Psi_{c}(x,t)=[-\frac{1}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{2}m\omega^{2}x^{2}]\Psi_{c}(x,t),$ (4.118)

with

$\Psi_{c}(x,t)=S^{-1}\Psi_{H.O}(x,t),$ (4.119)

where $\Psi_{c}(x,t)$ stands for the wave-function in commutative space-time.
Thus (4.118) is nothing but a time-dependent Schödinger equation for a one-
dimensional harmonic oscillator that lives in usual space-time.

Of particular interest are the stationary states, viz. eigenstates of the
Hamiltonian operator, which will obey the so called time independent
Schödinger equation:

$i\partial_{t}\Psi_{c}(x,t)=[-\frac{1}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{2}m\omega^{2}x^{2}]\Psi_{c}(x,t)=E\Psi_{c}(x,t)$ (4.120)

Following the usual approach, we now introduce the annihilation and creation
operators as

$b=\frac{1}{\sqrt{2m\omega}}(m\omega x+\frac{\partial}{\partial
x})~{};~{}b^{\dagger}=\frac{1}{\sqrt{2m\omega}}(m\omega
x-\frac{\partial}{\partial x})~{};~{}[b,b^{\dagger}]=1$ (4.121)

In terms of these operators we can re-write the above equation (4.120) as

$\omega\left(b^{\dagger}b+\frac{1}{2}\right)\Psi_{c}(x,t)=E\Psi_{c}(x,t)$
(4.122)

Since the operator $b^{\dagger}b$ is the number operator with non-negative
integers as its eigenvalues, the corresponding eigenvalues of the operator on
the left-hand side of the (4.122) gives the energy eigenvalues:
$E_{n}=\omega(n+\frac{1}{2})$. Therefore, the exactly solvable harmonic
oscillator model (4.118) in commutative space-time implies that the
corresponding model (4.113) in non-commutative space-time is also exactly
solvable, and vice versa. In fact, our models (4.113) and (4.118) in non-
commutative and commutative spaces, respectively, share the same set of energy
eigenvalues and their eigenstates relate to each other by the relation
(4.119).

The above discussions demonstrate that as long as the explicit $\theta$
dependence does not show up in the energy spectrum, it will be encoded only in
the stationary state wave-functions of harmonic oscillator 4.113 in non-
commutative space-time. To illustrate this point by an explicit example, let
us start by considering $\Psi^{(0)}_{c}(x,t)$, the ground state wave function
of (4.120) of commutative space-time with energy $E_{0}$, satisfying the
condition $b\Psi^{(0)}_{c}(x,t)=0$. Thus the time evolution of the _un-
normalized_ ground state wave-function factorizes as

$\Psi^{(0)}_{c}(x,t)=e^{-\frac{m\omega
x^{2}}{2}}e^{-iE_{0}t}=\int~{}dp~{}g^{(0)}(p)~{}e^{i(px-
E_{0}t)};~{}~{}~{}with~{}~{}g^{(0)}(p)=e^{-\frac{p^{2}}{2m\omega}}$ (4.123)

where $g^{0}(p)$ is a corresponding _un-normalized_ wave-function in momentum
space.

However, the corresponding ground state wave function in non commutative space
time i.e. in the coherent state representation ($\Psi^{(0)}_{H.O}(x,t)$) can
be easily obtained by making use of (4.114) and (4.119) to get the following
form of _un-normalised_ ground state wave function (normalizable with respect
to the inner product (4.87):

$\Psi^{(0)}_{H.O}(x,t)=S^{-1}\Psi^{(0)}_{c}(x,t)=e^{-\left[\frac{(x-\frac{\theta
E_{0}}{2})^{2}}{2\sigma^{2}_{\theta}}\right]}\,e^{-iE_{0}t}~{}~{};~{}\sigma^{2}_{\theta}=\frac{\theta}{2}+\frac{1}{m\omega}.$
(4.124)

This clearly displays energy ($E_{0}$) dependent shift in the origin towards
the right side and a modified width $\sigma_{\theta}$ which is a manifestation
of a breakdown of parity invariance. Clearly, this bears a resemblance to the
form of the ground state wave function in commutative space-time quantum
mechanics $\Psi^{0}_{c}(x,t)$, except for the $\theta$-deformation in the
width $\sigma_{\theta}$. With $\theta\to 0$, one gets back the familiar
commutative form of the _un-normalized_ ground state wave function (4.123).

Now we can compute the probability density factor $\rho_{\theta}$ using (4.83)
and integrate it over only $x$ to set it to unity. Thus, with the correct
normalization factor, we obtain

$\rho_{\theta}(x)=\frac{1}{\tilde{\sigma_{\theta}}\sqrt{2\pi}}e^{-\left[\frac{(x-\theta
E_{0})^{2}}{2\tilde{\sigma_{\theta}}^{2}}\right]}~{};~{}\tilde{\sigma_{\theta}}^{2}=\frac{\sigma_{\theta}^{2}}{2}\left[1+\frac{\theta}{2\sigma^{2}_{\theta}}\right]$
(4.125)

One can recast in a simpler form, using (4.67) as,

$\rho_{\theta}(x)=\delta_{\tilde{\sigma_{\theta}}}\left(x-\theta E_{0}\right)$
(4.126)

It may be noted the probability density-independent of time and is positive
definite. We can now study the effect of infinitely large confining potential,
by considering the limit $\omega\to\infty$ in presence of nonzero
noncommutativity $(\theta\neq 0)$. In this limit,
$\tilde{\sigma_{\theta}}\to\sqrt{\theta}$, preventing the squeezing of the
wave-packet in a region $\lesssim\sqrt{\theta}$. This is a purely a nontrivial
effect of noncommutativity as the inherent noncommutativity in space-time
provides a natural impenetrable barrier and does not allow for localization to
a sharply defined point. Thus the probability density $\rho(x)$ at some point
$x$ will have contributions from the vicinity and points from the finite,
however small, region around that point. In a certain sense, noncommutativity
therefore thus essentially introduces a non-locality within the theory, where
the notion of a point particle, it seems, has got to be necessarily replaced
by some extended object of a ‘cloud’ A similar situation was observed also in
the context of spatial noncommutativity [82].

Interestingly, if we takes both the limits $\omega\to\infty$ and
$\mid\theta\mid\to 0$, in such a way that their product is a finite constant:
$\omega\mid\sqrt{\theta}\mid\rightarrow~{}finite~{}constant$, then
$\rho(x)\to\delta(x)$, as one can expect in the commutative limit. One should
note that the the non-local features appear for both the ground state wave
function (4.124) and also in the expression of the probability density
(4.125). One can thus expect to see the effect of non-commutativity through
non-local behaviours in some measurable quantities or observables .

###  4.6 Transition probability in presence of space-time noncommutativity

In this section, we intend to investigate the modification due to the presence
of space-time noncommutativity, if any, in the rate of transition probability
amplitude of a system when it undergoes a transition from a given initial
stationary state to a particular final stationary state. For this study let us
first recall what we did for the case of time dependent perturbation theory in
ordinary quantum mechanics, described by a Hamiltonian $H_{0}$ and satisfies
time independent Schrödinger equation as

$H_{0}|\psi\rangle_{n}=E_{n}|\psi\rangle_{n}.$ (4.127)

These stationary eigenstates $|\psi\rangle_{n}$ evolve in time as

$|\psi(t)\rangle_{n}=e^{-iE_{n}t}|\psi\rangle_{n}.$ (4.128)

Now if we switch on a perturbation say $V(t)$ then its time evolution will be
governed by the Hamiltonian $H=H_{0}+V(t)$ satisfying time dependent
Schrödinger equation as

$H|\psi(t)\rangle=i\frac{\partial}{\partial t}|\psi(t)\rangle,$ (4.129)

We work with the ansatz for the perturbed state-vectors
$|\psi(t)\rangle=\sum_{n}C_{n}(t)e^{-iE_{n}t}|\psi\rangle_{n}$ where the
coefficients $C_{n}(t)$ are themselves explicitly time dependent.

In our non-commutative case, for physical states the above equation (4.129)
can be rewritten as

$H_{0}|\Psi;phy)+V(\hat{t})|\Psi;phy)=-P_{t}|\Psi;phy).$ (4.130)

Now using coherent state basis (4.56), above equation will take the following
form as

$(x,t|H_{0}|\Psi;phy)+(x,t|V(\hat{t})|\Psi;phy)=i\frac{\partial}{\partial
t}(x,t|\Psi;phy),$ (4.131)

or equivalently,

$H_{0}(P_{x},X_{\theta})\,\Psi_{phy}(x,t)+V(T_{\theta})\,\Psi_{phy}(x,t)=i\,\partial_{t}\Psi_{phy}(x,t).$
(4.132)

Let us now consider a generic non-stationary physical state which satisfies
the above Schrödinger equation and may be expanded as

$\Psi_{phy}(x,t)=\sum\limits_{n}C_{n}(t)e^{-iE_{n}t}\phi^{(n)}(x;\theta)$
(4.133)

with $H_{0}(P_{x},X_{\theta})\phi^{(n)}(x;\theta)=E_{n}\phi^{(n)}(x;\theta).$
Here the subscript $n$ implies to their association with the $n-th$ energy
levels $E_{n}$ of the unperturbed Hamiltonian $H_{0}$. Now we should insert
the expression of $\Psi_{phy}(x,t)$ in the above Schrödinger equation (4.132)
and solve with the initial condition $C_{n}(t=0)=\delta_{ni}$ (as the system
is in the initial state $\phi^{(i)}(x;\theta)$ at $t=0$). Proceeding thus we
have

$\begin{split}\sum_{n}i\dot{C}_{n}(t)\,&e^{-iE_{n}t}\phi_{n}(x;\theta)+\sum_{n}E_{n}C_{n}(t)\,e^{-iE_{n}t}\phi_{n}(x;\theta)\\\
&=\sum_{n}H_{0}(P_{x},X_{\theta})\,C_{n}(t)\,e^{-iE_{n}t}\phi_{n}(x;\theta)+\sum_{n}V(T_{\theta})\,C_{n}(t)\,e^{-iE_{n}t}\phi_{n}(x;\theta)\
\end{split}$ (4.134)

Simplifying,

$\sum_{n}i\dot{C}_{n}(t)\,e^{-iE_{n}t}\phi_{n}(x)=\sum_{n}V(T_{\theta})\,C_{n}(t)\,e^{-iE_{n}t}\phi_{n}(x).$
(4.135)

Now using (4.116) we obtain

$\sum_{n}i\dot{C}_{n}e^{-E_{n}t}\phi^{(n)}(x;\theta)=\sum_{n}S^{\dagger}V(t)(S^{\dagger})^{-1}C_{n}(t)e^{-iE_{n}t}\phi^{(n)}(x;\theta)$
(4.136)

where we have merely used the fact that
$V(S^{\dagger}t(S^{\dagger})^{-1})=S^{\dagger}V(t)(S^{\dagger})^{-1}.$

If the time-dependent perturbation $V(t)$ is weak then $C_{n}$for $n\neq i$
will also be a slowly varying function of time and its higher-order time
derivative can be disregarded to calculate the rate of transition probability.
At this stage, it is important to observe that the coherent state
representation of any bound stationary states
($e^{-iE_{n}t}\phi^{(n)}(x;\theta)$) in non-commutative space-time is related
to the stationary states ($e^{-iE_{n}t}\phi_{c}^{(n)}(x)$) of the
corresponding model in commutative space-time through the non-unitary operator
(4.114):

$e^{-iE_{n}t}\phi_{c}^{(n)}(x)=S^{-1}[e^{-iE_{n}t}\phi^{(n)}(x)]$ (4.137)

Now, after some algebra, the above equation (4.136) may be rewritten as

$\sum_{n}i\dot{C}_{n}~{}e^{-E_{n}t}\phi_{c}^{(n)}(x)=\sum_{n}C_{n}(t)~{}[UV(t)U^{-1}]~{}e^{-iE_{n}t}\phi_{c}^{(n)}(x).$
(4.138)

Therefore

$\sum_{n}i\dot{C}_{n}(t)~{}e^{-E_{n}t}\phi_{c}^{(n)}(x)=\sum_{n}C_{n}(t)~{}V(UtU^{-1})~{}e^{-iE_{n}t}\phi_{c}^{(n)}(x),$
(4.139)

with $U=e^{i\theta\partial_{x}\partial_{t}}.$ All information about space-time
noncommutativity may be encoded into this unitary operator $U$.

Finally, after using the Hadamard lemma, we arrive at

$\sum_{n}i\dot{C}_{n}(t)~{}e^{-E_{n}t}\phi_{c}^{(n)}(x)=\sum_{n}C_{n}(t)~{}V(t+i\theta\partial_{x})~{}e^{-iE_{n}t}\phi_{c}^{(n)}(x),$
(4.140)

Using a Taylor expansion up-to first order in $\theta$ (as $\theta$ is taken
to be very small and the perturbation is also weak), we can write,

$\begin{split}\sum_{n}i\dot{C}_{n}(t)\,e^{-iE_{n}t}\,\phi^{(n)}_{c}(x)&\simeq\sum_{n}V(t)\,C_{n}(t)e^{-iE_{n}t}\phi^{(n)}_{c}(x)\\\
&-\frac{\theta}{2}\sum_{n}C_{n}(t)\frac{\partial V(t)}{\partial
t}\left(-i\partial_{x}\right)e^{-iE_{n}t}\phi_{c}^{(n)}(x)+\mathcal{O}(\theta^{2})\
\end{split}$ (4.141)

Now taking the scalar product with the $e^{iE_{m}t}\,\phi^{*(m)}_{c}(x)$ we
arrive at

$\begin{split}i\dot{C}_{m}(t)&\simeq\sum_{n}\,C_{n}(t)\,e^{i(E_{m}-E_{n})t}\int~{}dx~{}\phi_{c}^{\ast(m)}(x)\,V(t)\,\,\phi_{c}^{(n)}(x)\\\
&-\frac{\theta}{2}\sum_{n}C_{n}(t)\,e^{i(E_{m}-E_{n})t}\int~{}dx~{}\phi_{c}^{\ast(m)}(x)\,\,\frac{\partial
V(t)}{\partial t}\,(-i\partial_{x})\,\,\phi_{c}^{(n)}(x)\ \end{split}$ (4.142)

Since the right-hand side of the above equation (4.142) has a matrix element
of $V$and it suffices to retain the term $C_{i}(t)$ there and for weak
perturbation $V(t)$ the coefficient $C_{i}(t)$ are approximately same as
$C_{i}(t=0)$ and $C_{i}$ too in this order of approximation is 1, and so
lowest contributing order $\theta$ is given by

$i\dot{C}_{m}(t)\simeq
e^{i(E_{m}-E_{n})t}[\,<\phi_{c}^{(m)}|V(t)|\phi_{c}^{(i)}>-\frac{\theta}{2}\langle\phi_{c}^{(m)}|\frac{\partial
V(t)}{\partial t}\hat{P}_{x}|\phi_{c}^{(i)}\rangle]$ (4.143)

Therefore if the system undergoes transition in time $T$, then for initial
state $|\phi_{c}^{(i)}\rangle$ and final state $|\phi_{c}^{(f)}\rangle$, we
have

$C_{f}(T)\simeq-i\int_{0}^{T}dt\,e^{i(E_{f}-E_{n})t}\left[<\phi_{c}^{(f)}|V(t)|\phi_{c}^{(i)}>-\frac{\theta}{2}\langle\phi_{c}^{(f)}|\frac{\partial
V(t)}{\partial t}\hat{P}_{x}|\phi_{c}^{(i)}\rangle\right]$ (4.144)

and the transition probability for the transition $i\rightarrow f$ is

$Prob_{f\leftarrow i}=\mid
C_{f}(T)\mid^{2}=\mid\int_{0}^{T}dt\,e^{i(E_{f}-E_{n})t}\left[<\phi_{c}^{(f)}|V(t)|\phi_{c}^{(i)}>-\frac{\theta}{2}\langle\phi_{c}^{(f)}|\frac{\partial
V(t)}{\partial t}\hat{P}_{x}|\phi_{c}^{(i)}\rangle\right]\mid^{2}$ (4.145)

The transition rate of the system for a total time $T$ is then given by

$\boldsymbol{Rate}_{f\leftarrow
i}=\frac{1}{T}\mid\int_{0}^{T}dt\,e^{i(E_{f}-E_{i})t}\left[<\phi_{c}^{(f)}|V(t)|\phi_{c}^{(i)}>-\frac{\theta}{2}\langle\phi_{c}^{(f)}|\frac{\partial
V(t)}{\partial t}\hat{P}_{x}|\phi_{c}^{(i)}\rangle\right]\mid^{2}$ (4.146)

In the smooth limit $\theta\rightarrow 0$, we get back the commutative result.
The explicit presence of $\theta$ dependent correction term in the relevant
first-order transition rate is clearly a non-commutative effect for which this
rate is found to get modified. Thus the decay of state due to the interaction
in non-commutative space-time is beautifully exhibited in this approach, and
this demonstration clearly indicates that the so-called Fermi-Golden rule has
to be modified in a nontrivial way.

###  4.7 Discussion

A toy-model of $(1+1)$-dimensional non-relativistic quantum mechanics on
quantum space-time have been studied in this chapter, wherein the quantum
structure of space-time arising from the canonical type of space-time non-
commutative algebra (4.45) which is relevant to capture the essential features
of the new quantum geometry of space-time. In fact, in Appendix A, using
Dirac’s constraint analysis, we obtain this non-commutative space-time
structure from a first-order non-relativistic system Lagrangian. It is
important to note that we did not fix any type of gauge to get the non-
commutative space-time algebra; this deformed non-commutative space-time
algebra was induced by a Chern-Simons like term in momentum space. Therefore,
in this sense, our method is different from some other approaches where
suitable gauge fixing was mandatory [218]. Then we show that by introducing
coherent state basis (4.56) it is possible to extract an effective commutative
theory starting from an operatorial version of Schrödinger equation (4.80)
using Hilbert-Schmidt operators. This equation now enjoys a similar form to
that of commutative quantum mechanics except that the point-wise product of
two functions is necessarily deformed by the Voros star product, ensuring
positive definiteness of probability density. Also since the star products
involve terms with infinite order derivatives of spatio-temporal variables,
this theory becomes highly non-local. We started from a time-re-
parametrization invariant form of action, wherein both space and time
coordinate are treated as a dynamical variables, in the sense that they are
both parts of an enlarged configuration space and eventually are promoted to
the level of quantum operators. However, the corresponding Hilbert space is
‘’bigger’ in the sense that we needed to introduce a space of physical states
that are equipped with the so-called ‘induced’ inner product to enable us to
provide the conventional probabilistic interpretation. And this is clearly
less stringent than the one used to define $L^{2}(R^{2})$.

Moreover, we have discussed how this formulation can be applied for the case
of free Gaussian wave packet and harmonic oscillator. For the free particle
case, we show explicitly that, the inherent noncommutativity does not allow to
localize the particle to a single point even if one introduces an infinite
uncertainty in the momentum space. We then compute harmonic oscillator
spectra, where we observed that the spectra remain same as we expect for a
usual oscillator. But, the corresponding energy eigenfunctions get deformed.
Specifically, the ground state wave functions indicate a parity-violating
shift in the origin and a modified width of the wave packet. We also show that
for an infinitely large confining potential, it is not possible to squeeze the
position of a particle under a threshold scale. This is certainly a non-local
feature of the theory arising purely from space-time noncommutativity. We
compute the rate of transition probability of a bound system under time-
dependent perturbation, which gives rise to a similar deformation in Fermi’s
golden rule.

###  4.8 Appendix A

Dirac procedure for generating the deformed space-time algebra

Starting with the first-order Lagrangian written below, here we are going to
carry out Dirac’s constraint Hamiltonian analysis and compute the Dirac’s
bracket among the configuration space variables where space-time and their
corresponding canonical momenta are considered to be configuration space
variables of extended phase-space as,

$L_{\theta}^{\tau}=p_{\mu}\dot{x}^{\mu}+\frac{\theta}{2}\epsilon^{\mu\nu}p_{\mu}\dot{p}_{\nu}-e(\tau)(p_{t}+H),\,\,\,\,\,\,\mu,\nu=0,1$
(4.147)

where $e(\tau)$ is an arbitrary Lagrange multiplier enforcing the constraint
$(p_{t}+H)\approx 0$ and the $\theta$ dependent term can be viewed as a Chern-
Simons term in the momentum space. Now we can derive the canonical momenta
corresponding to $x^{\mu},p_{\mu}$ and $e(\tau)$ as follows:

$\displaystyle\pi^{x}_{\mu}$ $\displaystyle=\frac{\partial
L_{\theta}^{\tau}}{\partial\dot{x}^{\mu}}=p_{\mu};$
$\displaystyle\pi_{p}^{\mu}$ $\displaystyle=\frac{\partial
L_{\theta}^{\tau}}{\partial\dot{p}_{\mu}}=-\frac{\theta}{2}\epsilon^{\mu\nu}p_{\nu};$
$\displaystyle\pi_{e}$ $\displaystyle=\frac{\partial
L_{\theta}^{\tau}}{\partial\dot{e}}=0$ (4.148)

We can see that the canonical momenta are not related to the generalized
velocities and so that the above equations (4.148) are interpreted as primary
constraints of the theory and can be written as

$\Phi_{1,{\mu}}=\pi_{\mu}^{x}-p_{\mu}\approx
0,\,\,\,\,\,\,\,\Phi_{2}^{\mu}=\pi^{\mu}_{p}+\frac{\theta}{2}\epsilon^{\mu\nu}p_{\nu}\approx
0,\,\,\,\,\,\,\,\Psi=\pi_{e}\approx 0$ (4.149)

Using the following Poission brackets between the canonical pairs,

$\\{x^{\mu},\pi_{\nu}^{x}\\}=\delta^{\mu}\,_{\nu},\,\,\,\,\,\,\\{p_{\mu},\pi^{\nu}_{p}\\}=\delta_{\mu}\,^{\nu};\\{e,\pi_{e}\\}=1$
(4.150)

we can derive the Poisson brackets between the constraints (4.149) as

$\displaystyle\\{\Psi,\Psi\\}=\\{\Psi,\Phi_{1,\mu}\\}=\\{\Psi,\Phi_{2}^{\mu}\\}=0$
$\displaystyle\\{\Phi_{1,{\mu}},\Phi_{1,{\nu}}\\}=0$
$\displaystyle\\{\Phi_{1,{\mu}},\Phi_{2}^{\nu}\\}=-\delta_{\mu}\,^{\nu}$
$\displaystyle\\{\Phi_{2}^{\mu},\Phi_{2}^{\nu}\\}=\theta\epsilon^{\mu\nu}$
(4.151)

The sign $\approx$ refers to weak equality in the sense of Dirac. By simple
Legendre transformation, we can obtain the canonical Hamiltonian as

$H_{c}=e(p_{t}+H)$ (4.152)

Since $\Psi$ has zero brackets with all other constraints, it can be
classified as a first-class constraint. The rest of the constraints are
classified as second class constraints as the Poisson’s bracket between of
$\Phi_{1,{\mu}}$ and $\Phi_{2,\mu}$ does not vanish. We can eliminate the
second class constraints by calculating the Dirac brackets.

Now the total Hamiltonian can be written by adding all primary constraints to
the canonical Hamiltonian as following

$H_{T}=e(\tau)(p_{t}+H)+\lambda_{1,\mu}\Phi_{1,\mu}+\lambda_{2,\mu}\Phi_{2}^{\mu}+\lambda\Psi$
(4.153)

where $\lambda_{1,\mu},\lambda_{2,\mu}$ and $\lambda$ are the suitable
Lagrange’s multipliers enforcing the respective constraints.. Now the
constraint should be zero at all times. So for a consistency check we are
going to derive the time derivative of the constraints using Hamiltonian
equation and check whether it gives zero; at least weakly as,

$\Phi:=\dot{\Psi}=\\{H_{T},\pi_{e}\\}=p_{t}+H\approx 0$ (4.154)

Hence, the consistency condition on the time derivative of $\Psi$ gives a
secondary constraint (4.154), i.e. the original constraint $\phi$ (4.20)
appears as a secondary constraint. No further new constraint is generated due
to the time conservation of secondary constraint (4.154) and the iterative
process thus terminates. The consistency condition of the other primary second
class constraints is used to fix the Lagrange multiplier
$\lambda_{1,\mu},\lambda_{2,\mu}$, which are not of our concern in this
context.

Before deriving the Dirac brackets, we need to write the constraint matrix and
its inverse, which are given below.

$\Lambda_{ab}=\begin{pmatrix}\\{\Phi_{1,{\mu}},\Phi_{1,{\nu}}\\}&\\{\Phi_{1,{\mu}},\Phi_{2}^{\nu}\\}\\\
\\{\Phi_{2}^{\nu},\Phi_{1,{\mu}}\\}&\\{\Phi_{2}^{\mu},\Phi_{2}^{\nu}\\}\end{pmatrix}=\begin{pmatrix}0&-\delta_{\mu}\,^{\nu}\\\
\delta^{\nu}\,_{\mu}&\theta\epsilon^{\mu\nu}\end{pmatrix}$ (4.155)
$(\Lambda^{-1})_{ab}=\begin{pmatrix}\theta\epsilon_{\mu\nu}&\delta^{\mu}\,_{\nu}\\\
-\delta_{\nu}\,^{\mu}&0\end{pmatrix}$ (4.156)

Note that $\Lambda_{ab}$ and $(\Lambda^{-1})_{ab}$ satisfy
$\Lambda_{ab}(\Lambda^{-1})_{bc}=\delta_{ac}$.

Knowing the inverse of the constraint matrix, we can easily calculate the
Dirac bracket between two phase-space variables as,

$\\{f,g\\}_{D}=\\{f,g\\}-\\{f,\Phi_{a}\\}(\Lambda^{-1})_{ab}\\{\Phi_{b},g\\}$
(4.157)

A straightforward computation then yields the relevant Dirac brackets between
the congiguration space variables as given below:

$\displaystyle\\{x^{\mu},x^{\nu}\\}_{D}=\theta\epsilon^{\mu\nu}$
$\displaystyle\\{p_{\mu},p_{\nu}\\}_{D}=0$
$\displaystyle\\{x^{\mu},p_{\nu}\\}_{D}=\delta^{\mu}\,_{\nu}$ (4.158)

So we can see that the Dirac bracket between the space-time variable gives a
non-zero result. We can now easily quantize the system by elevating the
classical variables to the level of operators and elevating the Dirac brackets
to the level of the commutator bracket.

We will also comment on the status of the primary first-class constraint
$\Psi$ and the secondary constraint $\Phi$. The primary first-class constraint
$\Psi$ (4.149) involves the canonical momentum corresponding to the Lagrange’s
multiplier $e(\tau)$, which is not physical in nature. So we can ignore this
constraint. On the other hand, if we compute the Dirac bracket between the
secondary constraint $\Phi$ (4.154), and all other primary constraints we will
get vanishing brackets. So $\Phi$ can be treated as a non-trivial first-class
constraint that eventually gives rise to the $\tau$ evolution of the system or
the generator of the gauge transformation of the theory.

###  4.9 Appendix B

Self adjoint property of quantum space-time operators ($X^{L}_{\theta}$,
$T^{L}_{\theta}$) on $\mathcal{A}_{\theta}$

Let us define the ’symbol’ mapping

$\displaystyle\mu:\mathcal{A}_{\theta}\rightarrow\mathbb{C},$
$\displaystyle\mu(\Psi(\hat{t},\hat{x}))=(z,\bar{z}|\Psi)=<z|\Psi(\hat{x},\hat{t})|z>$
(4.159)

Hence it can be shown that this is a homomorphism from
$\hat{\mathcal{A}}_{\theta}$ to $\mathbb{C}$ with the Voros star product
$\star_{V}$ viz.

$\mu(\Psi_{1}(\hat{x},\hat{t})\Psi_{2}(\hat{x},\hat{t})=(z,\bar{z}|\Psi_{1})\star_{V}(z,\bar{z}|\Psi_{2});~{}~{}|\Psi_{1}),|\Psi_{2})\in\hat{\mathcal{A}}_{\theta}$
(4.160)

As $\mu$ is onto and $\mu(I)=<z|I|z>=1$, this implies that $\mu$ is actually
an isomorphism.

We therefore have

$\mu(\Psi_{1}^{\dagger}(\hat{x},\hat{t})\hat{x}\Psi_{2}(\hat{x},\hat{t}))\rightarrow\Psi^{*}_{1}(x,t)\star_{V}(x\star_{V}\Psi_{2}(x,t))=\Psi^{*}_{1}\star_{V}X^{L}_{\theta}\Psi_{2}(x,t)$
(4.161)

Now the associativity of the Voros star product implies

$\Psi^{*}_{1}(x,t)\star_{V}(x\star_{V}\Psi_{2}(x,t))=(\Psi^{\star}_{1}(x,t)\star_{V}x)\star_{V}\Psi_{2}(x,t)$
(4.162)
$\implies\Psi^{*}_{1}\star_{V}X^{L}_{\theta}\Psi_{2}(x,t)=(X^{L}_{\theta}\Psi_{1}(x,t))^{*}\star_{V}\Psi_{2}(x,t)$
(4.163)

After integrating both sides with respect to $x$ we obtain

$\int dx~{}\Psi^{*}_{1}\star_{V}X^{L}_{\theta}\Psi_{2}(x,t)=\int
dx~{}(X^{L}_{\theta}\Psi_{1}(x,t))^{*}\star_{V}\Psi_{2}(x,t)$ (4.164)

Thus the operator $X^{L}_{\theta}$ is self adjoint with respect to induced
inner product (4.87). A similar calculation for the operator $T^{L}_{\theta}$
indeed gives rise to the self-adjoint nature for the induced inner product
itself.

##  Chapter 5 Conclusions

Even after several decades of intellectual investment, quantum gravity still
remain, arguably, the most important open problem in fundamental physics. The
problem of quantum gravity is being pursued through different approaches,
where the primary objective is to combine general relativity with quantum
mechanics. Although none of these theories are nowhere near their final stage
and many of their results remain tentative in nature, almost all these
competing approaches have one result in common and this is some form of
granular structure i.e. quantum space-time at Planck length scale, although
there is no consensus regarding the final mathematical structure of such
quantum space-time. For example, in the approach of loop quantum gravity
[219], the space-time was shown to be describable through certain spin-
network, where each node can be identified as “quantum of space”. Furthermore,
it was shown that area/ volume of a surface/ region are quantized in discrete
units and which are computable within the scope of the theory. On the other
hand, in the approach of non-commutative geometry, the structure of space-time
is taken to be non-commutative in nature, where space-time coordinates are
promoted to the level of operators, satisfying non-vanishing commutator
algebra. This can serve as possible deterrent against the possibility of
forming micro black holes, arising in any attempt to localize an event to this
scale and thereby effectively removing that part from the observable region.
This non-commutative structure of space-time was later corroborated by string
theory also, as a sort of low energy description.The main objective of this
thesis was to study different quantum theories on quantum space-time
manifested through its non-commutativity, i.e. of the latter type, taken in
its simplest form: the Moyal space/ space-time.

In the second chapter, we have shown how spatial noncommutativity emerges in a
planar quantum mechanical problem by taking the example of an interacting
electric dipole subjected to a constant magnetic field in the normal
direction. In this context, it is worth-mentioning that the parent dipole
model had particular applications in the context of fractional quantum Hall
effect (FQHE) for filling factor $\nu=\frac{1}{2}$ [220]. Here
noncommutativity appeared as an effective way of describing the interaction
with external (magnetic) field. Then we have adopted a Hilbert-Schmidt
operator formulation to provide a systematic quantum mechanical analysis of
the non-commutative model. A generating functional was constructed through a
coherent state path-integral approach. As we know, quantum mechanical models
may be interpreted as (0 + 1) dimensional field theory and can be regarded as
a precursor to genuine higher dimensional (i.e. $(2+1)$ or $(3+1)$\-
dimensional) field theories. Thus we could have interpreted our planar theory
by a 0 + 1 dimensional non-canonical complex scalar field theory. The
advantage of this perspective is that it facilitated the use of many of the
arsenals of QFT even in our quantum mechanical case. With this, we could
analyze the issues related to the time dilatation symmetry on non-commutative
spaces in the chapter. By exploiting the path-integral analysis, due to
Fujikawa, wherein the anomalous terms are identified with the Jacobian factors
obtained from the path-integral measure under dilatation transformation, we
showed that the Ward-Takahashi identities for such $0+1$ dimensional non-
commutative fields associated with broken scale invariance contain anomalies
in all orders.

In next chapter, we had studied emergence of the geometrical phase in a time-
dependent quantum mechanical system in non-commutative phase space. By making
use of proper non-canonical transformations we mapped the problem on effective
commutative space and determined the energy spectrum. Thereafter, exploiting
the adiabatic circuital theorem, we obtained an expression of geometrical
phase over and above the dynamical phase. Our findings agreed with the group-
theoretical analysis of Berry’s phase. As far as the physical origin of the
spatial non-commutative parameter is concerned, it can be identified with a
non-relativistic fractional spin in $2+1$ dimension. Given the recent findings
of anyonic excitations in some experiments [221], we believe our theoretical
results may become more relevant in explaining some of the observations in
anyonic systems in near future.

We then take up issues related to space-time noncommutativity in non-
relativistic quantum mechanics in chapter 4. Reperametrization invariant form
of $(1+1)$-dimensional action for a non-relativistic system is first
considered. Here we have shown, how non-commutative space-time can appear
naturally through the Dirac bracket of second class constraints, in presence
of a Chern-Simons like term in momentum space. Here Non-commutativity (NC) can
be interpreted as a fundamental constant of space-time. Then introducing
appropriate Hilbert space, a time-dependent Schrödinger equation in $1+1$
dimensional non-commutative space-time was obtained. A salient feature of our
approach is that there is no issue of violating unitarity. The time-
independent non-relativistic model of a free particle and a harmonic
oscillator living in this non-commutative space-time is then analysed. We
found that the spectrum of the system Hamiltonian remains unchanged, making it
virtually impossible to see any effect of noncommutativity in the dynamics of
the system. Rather, the wave function gets deformed. Additionally, for the
case of time-dependent perturbation, the non-commutative correction is shown
to appear as a perturbative expansion in the deformation parameter. By the
explicit calculation, we demonstrated that the leading order correction of
non-commutativity can have quite significant contribution in the rate of
transition amplitude.

Thus, we studied different aspects of non-commutative space-time in quantum
mechanics through the lens of $0+1$ dimensional nonlocal quantum field theory.
It was quite reassuring to see that in the limit of vanishing deformation
parameters the quantum theories reduced smoothly to their usual commutative
versions. In this way, we have successfully illustrated that if we revise the
notion of space-time at a short length scale by a non-commutative geometry
associated with one more deformation parameters, these could then be
considered as fundamental ingredients of a quantum theory of gravity.
Moreover, non-commutative quantum theories had many novel properties. In this
thesis, we have been strictly confined our attention to the planar quantum
systems. It would thus be an obvious suggestion for future research to
generalize this formalism to higher dimensions. As a future goal, we can
extend Hilbert Schmidt-operator formulation of non-commutative quantum
mechanics to the coherent state description of $\mathcal{P}$ representation of
mixed density matrix, which may admit an immediate hydrodynamics
interpretation [222] corresponding to the incompressible quantum fluid. On the
other hand, the use of multi-particle systems as a theoretical model for
searching some possible Planck scale manifestations has become a very
fascinating line of research [223, 224, 225]. Thus, our present investigation
may be the basis to develop the second quantized description of Schrödinger
field theory over non-commutative space-time through the Hilbert-Schmidt
operator approach, which can perhaps provide a natural background to obtain
possible signals caused by the quantum fluctuation of space-time in low-energy
earth-based experiments. Furthermore, quantum space-time structure arising
from the relevant nontrivial commutator of $[\hat{t},\hat{x}]$ also may be
extended to understand the structure of quantum Minkowski, Rindler, and
Schwarzschild space-times and study their implications.

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#  Early excess emission in Type Ia supernovae from the interaction between
supernova ejecta and their circumstellar wind

Takashi J. Moriya,1,2 Paolo A. Mazzali,3,4 Chris Ashall,5 and Elena Pian6
1National Astronomical Observatory of Japan, National Institutes of Natural
Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2School of Physics and Astronomy, Faculty of Science, Monash University,
Clayton, Victoria 3800, Australia
3Astrophysics Research Institute, Liverpool John Moores University, IC2,
Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK
4Max-Planck Institute for Astrophysics, Karl-Schwarzschild-Straße 1, 85748
Garching, Germany
5Department of Physics, Virginia Tech, 850 West Campus Drive, Blacksburg VA,
24061, USA
6INAF, Astrophysics and Space Science Observatory, via P. Gobetti 101, 40129
Bologna, Italy
E-mail<EMAIL_ADDRESS>(TJM)

(Accepted 2023 May 04. Received 2023 April 15; in original form 2023 February
08)

###### Abstract

The effects of the interaction between Type Ia supernova ejecta and their
circumstellar wind on the photometric properties of Type Ia supernovae are
investigated. We assume that a hydrogen-rich, dense, and extended
circumstellar matter (CSM) is formed by the steady mass loss of their
progenitor systems. The CSM density is assumed to be proportional to $r^{-2}$.
When the mass-loss rate is above
$10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ with a wind velocity of 100
$\mathrm{km~{}s^{-1}}$, CSM interaction results in an early flux excess in
optical light-curves within 4 days of explosion. In these cases, the optical
colour quickly evolves to the blue. The ultraviolet flux below 3000 Å is found
to have a persistent flux excess compared to Type Ia supernovae as long as CSM
interaction continues. Type Ia supernovae with progenitor mass-loss rates
between $10^{-4}$ and $10^{-3}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ may
not have a CSM that is dense enough to affect spectra to make them Type Ia-
CSM, but they may still result in Type Ia supernovae with an early optical
flux excess. Because they have a persistent ultraviolet flux excess,
ultraviolet light curves around the luminosity peak would be significantly
different from those with a low-density CSM.

###### keywords:

supernovae: general – circumstellar matter

††pubyear: 2023††pagerange:  Early excess emission in Type Ia supernovae from
the interaction between supernova ejecta and their circumstellar wind – Early
excess emission in Type Ia supernovae from the interaction between supernova
ejecta and their circumstellar wind

##  1 Introduction

Type Ia supernovae (SNe Ia) are thermonuclear explosions of white dwarfs
(e.g., Nugent et al., 2011; Bloom et al., 2012). The evolutionary path leading
to white dwarf explosions is a long-standing issue in astrophysics (e.g.,
Maeda & Terada, 2016; Livio & Mazzali, 2018; Wang, 2018; Soker, 2019, for
recent reviews). Two major paths have been proposed. One is a so-called
“single-degenerate” channel in which a white dwarf gains its mass through the
accretion from a non-degenerate companion star (e.g., Nomoto, 1982). The other
path is a “double-degenerate” channel in which coalescence of binary white
dwarfs leads to a thermonuclear explosion (e.g., Iben & Tutukov, 1984;
Webbink, 1984).

The two channels predict different properties of circumstellar matter (CSM)
around SNe Ia. In the case of the single-degenerate channel, mass transfer
from a non-degenerate star to a white dwarf can lead to mass loss (e.g.,
Chomiuk et al., 2012; Moriya et al., 2019, and references therein). Thus,
single-degenerate systems can lead to the formation of an extended dense
hydrogen-rich or helium CSM (e.g., Dragulin & Hoeflich, 2016). However, the
double-degenerate channel involves two white dwarfs. Therefore, no significant
mass loss forming a dense extended CSM is expected from such a system before
the explosion. Observationally, a fraction of SNe Ia show clear signatures of
interaction between the SN Ia ejecta and an extended dense hydrogen-rich or
helium CSM in their optical light-curves and spectra (e.g., Hamuy et al.,
2003; Dilday et al., 2012; Silverman et al., 2013; Kool et al., 2022; Sharma
et al., 2023), and they are likely related to the single-degenerate channel.
It is also possible that SN Ia signatures are often hidden in the interaction
features and a certain fraction of interacting SNe are linked to SNe Ia (e.g.,
Leloudas et al., 2015; Jerkstrand et al., 2020). The fact that some SNe Ia
tend to appear in star-forming regions also suggests importance of large mass
loss in their progenitor systems and implies the existence of dense CSM around
SNe Ia (e.g., Bartunov et al., 1994; Anderson et al., 2015; Pavlyuk &
Tsvetkov, 2016; Hakobyan et al., 2016).

Several observational features have been linked to the possible presence of
hydrogen-rich dense CSM around SNe Ia. For example, strong calcium absorption
observed in early spectra of SNe Ia has been related to the existence of
hydrogen-rich CSM (e.g., Gerardy et al., 2004; Mazzali et al., 2005a, b;
Tanaka et al., 2008). Some SNe Ia also show hydrogen emission lines at late
phases that could be excited by CSM interaction (e.g., Kollmeier et al., 2019;
Vallely et al., 2019; Prieto et al., 2020; Elias-Rosa et al., 2021). In
addition, narrow, blueshifted sodium absorption lines detected in SNe Ia are
suggested to indicate the existence of extended CSM around their progenitors
(e.g., Patat et al., 2007; Maguire et al., 2013). The density of extended CSM
around SN Ia progenitor systems can also be probed by radio and X-ray
observations of SNe Ia. However, no radio and X-ray emission from SNe Ia has
been observed (e.g., Chomiuk et al., 2012, 2016; Pérez-Torres et al., 2014;
Harris et al., 2016, 2018, 2021; Lundqvist et al., 2020; Russell & Immler,
2012; Margutti et al., 2012, 2014; Dwarkadas, 2023) except for SN 2020eyj
(Kool et al., 2022)111There is a controversial case of SN 2005ke (Immler et
al., 2006; Hughes et al., 2007). SN 2012ca was detected in X-rays (Bochenek et
al., 2018), but it may not be a SN Ia (Inserra et al., 2016).. The lack of
radio and X-ray detection indicates that the CSM density around SNe Ia is
generally not as high as those predicted by the single-degenerate channel, but
it is important to note that the inferred CSM density depends on uncertain
physics determining the emission properties (e.g., Chomiuk et al., 2016). It
is also possible that there is a delay between the accretion phase and
explosion in the single-degenerate channel (e.g., Di Stefano et al., 2011;
Justham, 2011).

In addition to the aforementioned CSM signatures, early emission from SNe Ia
is suggested to contain information on progenitor systems of SNe Ia. Recent
early SN Ia observations are starting to reveal that some SNe Ia have excess
optical emission within several days after the explosions (e.g., Goobar et
al., 2014; Cao et al., 2015; Marion et al., 2016; Jiang et al., 2017; Jiang et
al., 2018; Jiang et al., 2020, 2021; Hosseinzadeh et al., 2017, 2022; Miller
et al., 2018, 2020; Shappee et al., 2019; Li et al., 2019; Dimitriadis et al.,
2019; Tucker et al., 2021; Burke et al., 2021; Ashall et al., 2022; Sai et
al., 2022; Ni et al., 2022; Srivastav et al., 2022; Lim et al., 2023). The
fraction of SNe Ia showing early excess emission is estimated to be $\sim
20-30$% (e.g., Deckers et al., 2022; Magee et al., 2022). Early excess
emission was originally suggested to be caused by the impact of the SN ejecta
on a non-degenerate companion star in the progenitor system, and therefore to
constitute evidence in favour of the single-degenerate channel (e.g., Kasen,
2010; Kutsuna & Shigeyama, 2015; Maeda et al., 2014; Liu et al., 2015). Other
mechanisms have been proposed to cause early excess emission, such as mixing
of ${}^{56}\mathrm{Ni}$ to outer layers (e.g., Piro & Morozova, 2016; Noebauer
et al., 2017; Maeda et al., 2018; Polin et al., 2019; Magee & Maguire, 2020;
Magee et al., 2021), Doppler line shifts (Ashall et al., 2022), and existence
of carbon and oxygen-rich envelope around white dwarfs (e.g., Ashall et al.,
2021; Maeda et al., 2023). In addition to the excess emission, early colour
evolution is found to have diversity in SNe Ia (e.g., Stritzinger et al.,
2018; Bulla et al., 2020; Burke et al., 2022).

Another possible way to produce early excess emission is the interaction
between the SN ejecta and a locally produced CSM. Interaction between SN
ejecta and a confined, dense CSM composed mostly of carbon and oxygen has been
suggested to cause early excess emission in SNe Ia (e.g., Dessart et al.,
2014a; Levanon et al., 2015; Levanon & Soker, 2017, 2019; Piro & Morozova,
2016; Kromer et al., 2016). This type of CSM is expected to form during the
merger of two white dwarfs in the double-degenerate channel (e.g., Hachinger
et al., 2012; Schwab et al., 2012; Shen et al., 2012; Pakmor et al., 2013;
Tanikawa et al., 2015). However, in the case of the single-degenerate channel,
extended hydrogen-rich CSM formed by the mass loss from progenitor systems is
expected to exist as discussed earlier. In this study, we investigate how the
early emission properties of SNe Ia are affected by the interaction between SN
ejecta and such an extended hydrogen-rich CSM.

The rest of the paper is organized as follows. We first show our numerical
modelling setups and the initial conditions of our numerical simulations in
Section 2. We summarize our results in Section 3. We discuss our results in
Section 4 and conclude this paper in Section 5.

##  2 Methods

###  2.1 Numerical modelling

The interaction between SN ejecta and CSM involves the conversion of kinetic
energy of SN ejecta to radiation. Thus, it is necessary to treat hydrodynamics
and radiation transfer, as well as their mutual effects, simultaneously. We
use the one-dimensional multi-frequency radiation hydrodynamics code STELLA
(Blinnikov et al., 1998, 2000, 2006) to investigate the consequences of the
interaction between SN ejecta and CSM in SNe Ia. STELLA has been used for
light-curve modelling of SNe Ia (e.g., Blinnikov et al., 2006; Woosley et al.,
2007; Kamiya et al., 2012; Noebauer et al., 2017), including the interaction
between SN Ia ejecta and a confined, dense carbon-oxygen CSM (Noebauer et al.,
2016). In short, STELLA numerically treats the hydrodynamic equations coupled
with radiation, and implicitly solves the time-dependent equations of the
angular moments of the radiation intensity averaged over a frequency bin by
using the variable Eddington method. STELLA calculates spectral energy
distributions (SEDs) at each time step. Multicolour light-curves can be
obtained by convolving filter functions with the SEDs. The standard 100
frequency bins in the SED calculations, i.e., from 1 Å to $5\times 10^{4}$ Å
on a log scale, are adopted in this study. STELLA assumes spherical symmetry
in calculating hydrodynamic and radiation properties. Although SNe Ia and
their CSM may not be spherical (e.g., Liu et al., 2017), our assumption of
spherical symmetry would be good enough to obtain the overall properties of
CSM interaction features in SNe Ia (cf. Cikota et al., 2019).

Figure 1:  Initial density structure in our light-curve calculations. The CSM
with $\rho_{\mathrm{CSM}}\propto r^{-2}$ are attached on top of the W7 density
structure at $4\times 10^{13}~{}\mathrm{cm}$. The mass-loss rate of each
model, as well as the mass in each CSM, is shown in the figure. We assume a
wind velocity of 100 $\mathrm{km~{}s^{-1}}$.

Figure 2:  Bolometric (top) and pseudo-bolometric (bottom) light-curves
obtained from our numerical calculations. The bolometric light-curves include
the flux in all the wavelengths and the pseudo-bolometric light-curves are
obtained by integrating the optical flux from 3250 Å to 8900 Å.

Figure 3:  Same as Fig. 2, but we additionally show the models with different
CSM radii in the case of $\dot{M}=5\times
10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$.

###  2.2 Initial conditions

We take the hydrodynamic structure of the W7 model in Nomoto et al. (1984);
Thielemann et al. (1986) to represent the SN Ia ejecta. The W7 model is a
carbon fast deflagration explosion model of a white dwarf. The W7 model is
known to explain the overall properties of SNe Ia (e.g., Nugent et al., 1997;
Tanaka et al., 2011), although its nucleosynthesis is known to have some
issues (e.g., Iwamoto et al., 1999). Because in the W7 model the SN ejecta
properties and the ${}^{56}\mathrm{Ni}$ mass, which are the most influential
photometric properties of SNe, are overall consistent with SN Ia observations
(e.g., Stehle et al., 2005; Mazzali et al., 2008), it is reasonable to adopt
it to investigate the effects of CSM interaction in SN Ia photometric
properties.

Before colliding on the CSM, the SN Ia ejecta are assumed to expand
homologously for $2\times 10^{4}~{}\mathrm{sec}\simeq 0.2~{}\mathrm{days}$
when the outermost layers reach $4\times 10^{13}~{}\mathrm{cm}$. Because the
most extended companion stars in SN Ia progenitor systems are likely red
giants having radii of $\sim 10^{13}~{}\mathrm{cm}$ (e.g., Hachisu et al.,
1996), the CSM interaction may actually start earlier. Still, the collision
time is small compared to the timescales of significant changes in
photospheric properties. These occur on a timescale of several days, as shown
below, such that the assumed collision radius is reasonable.

The CSM structure is attached to the SN ejecta at $4\times
10^{13}~{}\mathrm{cm}$ and extends above. If we assume that the progenitor
system experienced steady mass loss, the CSM density can be expressed as

$\rho_{\mathrm{CSM}}=\frac{\dot{M}}{4\pi v_{\mathrm{wind}}}r^{-2},$ (1)

where $\dot{M}$ is the mass-loss rate, $v_{\mathrm{wind}}$ is the wind
velocity, and $r$ is the radius. The CSM density, which is the most
influential parameter in CSM interaction, is determined by
$\dot{M}/v_{\mathrm{CSM}}$. In this paper, we assume
$v_{\mathrm{wind}}=100~{}\mathrm{km~{}s^{-1}}$ and vary $\dot{M}$ to
investigate the effects of the interaction between SN Ia ejecta and CSM. Our
results are not significantly affected by the wind velocity as long as this is
smaller than the typical SN Ia ejecta velocity which is much larger than
$1000~{}\mathrm{km~{}s^{-1}}$. Wind velocities of SN Ia progenitors are
expected to be $\sim 10-1000~{}\mathrm{km~{}s^{-1}}$ (e.g., Chomiuk et al.,
2016). The CSM extends to $10^{16}~{}\mathrm{cm}$ in our standard models, so
that the interaction continues through the peak of the light-curve. The choice
of CSM radius does not affect our conclusions as long as the radius is
sufficiently large, as discussed below. We also show some models with
different CSM radii. The CSM is assumed to have solar composition. The exact
composition, however, does not affect our results because the emission
properties are mostly determined by the hydrodynamics. Fig. 1 shows the
initial density structure of our standard models. Our models cover from
$\dot{M}=10^{-5}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ to
$10^{-3}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$.

The major differences between our initial conditions and those in previous
studies of interaction between SN Ia ejecta and CSM concern the density
structure. Previous studies considered a CSM formed in the mergers of two
white dwarfs and adopt a CSM density structure $\rho_{\mathrm{CSM}}\propto
r^{-3}$ (e.g., Piro & Morozova, 2016; Maeda et al., 2018). Thus, the radius of
the dense CSM in those studies is smaller than in our model. The CSM
composition is also different, but this is not expected to affect interaction-
powered light curves, as they are powered mechanically.

Figure 4:  Optical light-curves obtained from our numerical calculations.

Figure 5:  Ultraviolet light-curves obtained from our numerical calculations.

##  3 Results

###  3.1 Bolometric light-curves

The top panel of Fig. 2 presents our synthetic bolometric light-curves. We can
find that the existence of CSM affects light-curve properties of SNe Ia. When
$\dot{M}\lesssim 2.5\times 10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$,
the bolometric luminosity of SN Ia models with CSM deviate from that without
CSM mostly at the earliest phases, within 4 days of explosion. Although CSM
interaction continues at later phases, the luminosity from CSM interaction is
much less than the luminosity caused by heating from ${}^{56}\mathrm{Ni}$ and
${}^{56}\mathrm{Co}$ decay. When $\dot{M}\gtrsim 5\times
10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$, the luminosity provided by
CSM interaction becomes sufficiently high and the bolometric luminosity is
higher than that in the model without CSM as long as the interaction
continues. Even in these cases, tthough, the contribution of CSM impact to the
luminosity is most significant at the earliest phases, within 4 days of
explosion.

The bottom panel of Fig. 2 shows the pseudo-bolometric luminosity evolution
that is obtained by integrating the fluxes at optical wavelengths ($3250-8900$
Å). Compared to the bolometric light-curves, excess luminosity due to CSM
interaction is only noticeable within 4 days of explosion in all the models.
As discussed in the next section, the luminosity excess at later epochs is
mostly at ultraviolet (UV) wavelengths. The effect of CSM interaction is only
visible at optical wavelengths when $\dot{M}\gtrsim
10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$, and only in the earliest
phases.

Fig. 3 shows the effects of CSM radius on the bolometric and pseudo-bolometric
light-curves. The CSM is artificially cut at $10^{16}~{}\mathrm{cm}$ in our
models, but the actual outer edge of the CSM is determined by the duration of
the mass-loss phase from the progenitor system. Although there are slight
differences caused by the CSM radius, the effect of CSM density is found to be
much more significant than that of CSM radius. When the CSM radius is too
small, however, CSM interaction does not persist long enough for a luminosity
excess to be observed. To summarize, our conclusions in this work are not
affected by our assumption on the CSM radius if this is large enough to cause
CSM interaction at the earliest phases.

###  3.2 Optical and ultraviolet light-curves

Fig. 4 shows our synthetic light-curves in optical bands. We show the light-
curves in the B and V bands from the Carnegie Supernova Project (CSP, Hamuy et
al. 2006) and the g and r bands from the Zwicky Transient Facility (ZTF, Bellm
et al. 2019). As discussed in the previous section, optical light-curves are
only affected at the earliest phases within 4 days of explosion. We can find
that CSM interaction results in a significant early-phase flux excess in the
optical bands when $\dot{M}\gtrsim
10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$.

The light-curve evolution in UV bands below 3000 Å is presented in Fig. 5. We
adopt the u band ($3000-4000$ Å) from CSP, and the uvw2 and uvw1 bands from
Neil Gehrels Swift Observatory (Gehrels et al., 2004; Roming et al., 2005),
which are sensitive below 3000 Å. The u band light-curves have similar
properties to those of the optical light-curves with the early-phase excess
emission discussed above. However, UV light-curves below 3000 Å show
significantly different light-curve features from those found in optical
light-curves. In particular, the UV light-curves computed using
$\dot{M}\gtrsim 5\times 10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ show
a persistent flux excess at all phases when interaction is active. The SN UV
flux below 3000 Å is usually suppressed in SNe Ia by absorption by Fe-group
elements (e.g., Höflich et al., 1998; Mazzali, 2000; Lentz et al., 2000; Kasen
et al., 2006; Mazzali et al., 2014; Dessart et al., 2014b). Meanwhile, a large
fraction of the radiation from CSM interaction is emitted at the UV
wavelengths because of the high shock temperature ($\sim 10^{4}$ K, e.g.,
Moriya et al. 2011). This UV flux is emitted at large radii and can therefore
escape. Thus, the effect of CSM interaction is seen mainly at UV wavelengths,
and the extra flux from CSM interaction can easily dominate the UV flux of
interacting SNe Ia.

###  3.3 Colour

Fig. 6 shows that the colour evolution of our synthetic SN Ia models is also
affected by CSM interaction. When the CSM density is high, SNe Ia are found to
become blue more rapidly because of stronger CSM interaction. We discuss a
comparison with observations in the next section.

Figure 7 compares the colour evolution of several models predicting early flux
excess in SNe Ia from Maeda et al. (2018). A CSM interaction model with a CSM
density distribution of $\rho_{\mathrm{CSM}}\propto r^{-3}$ from a WD merger,
a companion interaction model, and three helium detonation models
(${}^{56}\mathrm{Ni}$ at outer layers) are compared. Our CSM interaction model
($\rho_{\mathrm{CSM}}\propto r^{-2}$) is found to have a distinct colour
evolution compared to other models.

Figure 6:  Colour evolution of our synthetic light-curves. The top panel
presents the $B-V$ colour evolution and the bottom panel presents the $g-r$
colour evolution. We also show the observed colour evolution of SNe Ia from
Stritzinger et al. (2018, top) and Bulla et al. (2020, bottom). In the top
panel, each SN Ia is plotted with the same symbol as in Stritzinger et al.
(2018), and the colour of the symbols is based on the early-phase behavior as
defined in Stritzinger et al. (2018). The green symbol is the intermediate
case. The details of each SN Ia can be found in Stritzinger et al. (2018). SN
2017cbv is shown by blue circles.  Figure 7:  Comparisons of the $B-V$ colour
evolution among different theoretical models predicting early flux excess in
SNe Ia. Representative models in Maeda et al. (2018) are presented.

Figure 8:  Scaled B band light-curves of the SNe Ia shown in the top panel of
Fig. 6, which are taken from Stritzinger et al. (2018). The observed
magnitudes are scaled to match the synthetic models at light-curve peak. The
top panel shows the light-curves until 30 days after explosion and the bottom
panel shows the light-curves until 10 days after explosion. SN 2017cbv is
shown as blue circles.

##  4 Discussion

We have shown that the collision between SNe Ia ejecta and an extended CSM
caused by a steady wind from the progenitor system leads to a flux excess in
the SN optical light-curves in the first 4 days after explosion if
$\dot{M}\gtrsim 10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ with a wind
velocity of 100 $\mathrm{km~{}s^{-1}}$ (Fig. 4). The colour evolution is also
affected by CSM interaction (Fig. 6). It has been suggested that there may be
distinct populations in colour evolution in the early phases ($\gtrsim
4~{}\mathrm{days}$) of SNe Ia (Stritzinger et al., 2018), although some
studies find the evidence less conclusive (e.g., Bulla et al., 2020). In order
to test whether interaction with a CSM could lead to the observed behaviour,
the observed colour evolution of SNe Ia is compared to the colour evolution in
our synthetic models in Fig. 6. In the case of the $B-V$ colour evolution
presented by Stritzinger et al. (2018), we find that SNe Ia with an early red
colour are consistent with the models with $\dot{M}\lesssim
10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$, while SNe Ia with an early
blue colour evolution are more consistent with models with $\dot{M}\gtrsim
2.5\times 10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$. The change from
red evolution to blue evolution occurs over a small range of $\dot{M}$
($10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}\lesssim\dot{M}\lesssim
2.5\times 10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$).

The ZTF SNe Ia sample (Bulla et al., 2020) does not show the presence of two
groups with different $g-r$ colour evolution (Fig. 6). Our synthetic models
match the $B-V$ evolution of the SNe Ia sample in Stritzinger et al. (2018)
well, from early to late phases, but the $g-r$ colour evolution is quite
different at most phases from that of the ZTF SNe Ia as reported by Bulla et
al. (2020). We cannot offer a clear reason for this difference. As our
numerical calculations do not resolve spectral line features in the g and r
bands well, some lines in spectra might be responsible for the difference. It
is also possible that the explosion dates of the ZTF SN Ia sample, which are
estimated by adopting a power-law function, are not well determined, and there
might be a systematic shift in the colour evolution.

If the separation in $B-V$ colour evolution at early phases is the result of a
difference in CSM, SNe Ia that are blue at the early phases should always have
an early flux excess from CSM interaction. Fig. 8 shows the $B$ band light-
curves of SNe Ia in the Stritzinger’s sample (top panel of Fig. 6). SNe Ia
with an early B-band flux excess tend to have blue colour. In particular, the
flux excess and early colour evolution of SN 2017cbv (Hosseinzadeh et al.,
2017; Stritzinger et al., 2018) are well explained by our CSM interaction
model. However, one SN Ia (SN 2009ig, Blondin et al. 2012; Foley et al. 2012;
Marion et al. 2013) with early flux excess has red colour evolution. This
suggests that more than one mechanism may be responsible for the early flux
excess, as discussed in Section 1. The colour evolution of SN 2017cbv towards
the blue is faster than the model prediction. It may be an effect of
asphericity in the CSM, allowing photons from the interaction region to escape
faster. The discrepancy between the synthetic and observed light curves after
peak seen in Fig. 8 may be caused by differences inside the ejecta and it does
not affect the early bumps caused by the interaction with the outermost
ejecta.

An early flux excess due to CSM interaction requires mass-loss rates from SN
Ia progenitor systems of at least $\dot{M}\gtrsim
10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$, assuming a wind velocity of
100 $\mathrm{km~{}s^{-1}}$. Such high mass-loss rates are usually excluded by
radio and X-ray observations (Section 1). However, we do know that some SN Ia-
CSM progenitor systems had mass-loss rates exceeding
$10^{-3}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ (e.g., Sharma et al., 2023).
It is possible that SN Ia progenitor systems with
$10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}\lesssim\dot{M}\lesssim
10^{-3}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ do not exhibit clear CSM
interaction signatures in spectra as do SNe Ia-CSM, but only show a flux
excess in the light-curves.

SNe Ia with a moderate mass-loss rate such that they only show the early flux
excess in optical bands but no spectroscopic signature may still show
significant, persistent flux excess also in the UV, and CSM interaction may
thus explain some diversity in the UV properties of SNe Ia (e.g., Brown et
al., 2012; Brown & Crumpler, 2020; Pan et al., 2018; Pan et al., 2020;
Srivastav et al., 2022; Sauer et al., 2008). Because $20-30$% of SNe Ia show
early flux excess (e.g., Deckers et al., 2022; Magee et al., 2022) and there
are several mechanisms to cause early flux excess, up to around 10% of SNe Ia
may have such early flux excess due to CSM interaction discussed in this
paper. Chomiuk et al. (2016) estimated that less than 10% of SNe Ia have mass-
loss rates higher than $10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$ based
on radio luminosity limits. This is consistent with our estimates of the
fraction of SNe Ia with early flux excess from wind interaction. SN Ia
progenitor systems with high mass-loss rates may result from optically thick
winds (e.g., Kato & Hachisu, 1994) or common-envelope winds (e.g., Cui et al.,
2022), for example. The fraction of SNe Ia with early flux excess from the CSM
interaction is presumably higher than the fraction of SNe Ia-CSM among SNe Ia
($0.02-0.2$% of SNe Ia, Sharma et al. 2023).

Finally, we note that SNe Ia without early flux excess show a rather slow
light-curve rise, often referred to as a “dark phase” (e.g., Mazzali et al.,
2001; Piro & Nakar, 2013). Fitting bolometric light curves with a $t^{2}$
function leads to an explosion date estimate of about one day later than the
actual explosion date (Fig. 2). A difference of even just one day may be
critical in interpreting the earliest photometric and spectroscopic properties
of SNe Ia. Interpretation of early observational data needs a careful
assessment of the explosion date.

##  5 Conclusions

We investigated the effect on the photometric properties of SNe Ia of the
interaction between the SN ejecta and a surrounding hydrogen-rich, dense,
extended CSM. The CSM is assumed to be formed by steady mass loss from the
progenitor system. It is assumed to have an extended structure with
$\rho_{\mathrm{CSM}}\propto r^{-2}$ (Eq. 1) and solar metallicity. An early
flux excess in optical light-curves is caused by CSM interaction if
$\dot{M}\gtrsim 10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$. UV (below
3000 Å) light-curves are affected significantly and the flux excess there
persists as long as CSM interaction continues.

CSM interaction also affects the early colour evolution in SNe Ia. When the
CSM density is high enough to cause an early optical flux excess, the optical
colour is also found to evolve rapidly towards the blue. Thus, CSM interaction
may be partly responsible for the diversity of the early colour evolution that
has been seen in SNe Ia. It is possible that a fraction of SNe Ia do not have
high enough CSM density for the spectra to display the features of SNe Ia-CSM
($\dot{M}\gtrsim 10^{-3}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$), but still
have a sufficient CSM density to cause early-phase flux excess in optical
light-curves. This requires
($10^{-4}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}\lesssim\dot{M}\lesssim
10^{-3}~{}\mathrm{M}_{\odot}~{}\mathrm{yr^{-1}}$). CSM interaction should also
affect the X-ray and radio properties. Multi-wavelength observations of SNe Ia
is therefore important in constraining their CSM properties. Such high mass-
loss rates are rather extreme even among single-degenerate progenitor systems.
However, even though systems with such high mass-loss rates may be rare, they
can provide important clues as to the evolutionary path towards SNe Ia.
Because the CSM interaction signatures appear clearly in the UV, early SN Ia
observations by future ultraviolet satellites such as
ULTRASAT222https://www.weizmann.ac.il/ultrasat/ and
CASTOR333https://www.castormission.org/ will be critical in uncovering the
circumstellar environments and progenitor systems of SNe Ia.

## Acknowledgements

We thank Benjamin Shappee and Maximilian Stritzinger for useful discussions.
This work was supported by the NAOJ Research Coordination Committee, NINS
(NAOJ-RCC-22FS-0502, NAOJ-RCC-22FS-0503). PAM and EP are grateful for support
and hospitality at NAOJ in Mitaka under program NAOJ-RCC-22FS-0502 and NAOJ-
RCC-22FS-0503. TJM is supported by the Grants-in-Aid for Scientific Research
of the Japan Society for the Promotion of Science (JP20H00174, JP21K13966,
JP21H04997). AC acknowledges support by NASA grant JWST-GO-02114.032-A.
Numerical computations were in part carried out on PC cluster at Center for
Computational Astrophysics (CfCA), National Astronomical Observatory of Japan.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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