Path Integral in Modular Space

The Schrödinger representation is based on a commutative subalgebra of $\mathcal{W}$ that is spanned by the elements $\{{\hat{W}_{(0,b)}}\,|\,b\in{\mathbb{R}}^{d}\}$. In other words, the position operators $\hat{q}^{a}$ and their exponentials are diagonalized in this representation. Their common eigenvectors are denoted by $\ket{x}_{\mathrm{Sch}}$, $x\in{\mathbb{R}}^{d}$, and they satisfy $\hat{q}^{a}\ket{x}_{\mathrm{Sch}}=x^{a}\ket{x}_{\mathrm{Sch}}$ and ${\hat{W}_{(0,b)}}\ket{x}_{\mathrm{Sch}}=e^{ib\cdot x/\hbar}\ket{x}_{\mathrm{Sch}}$.

A general quantum state can be written in the Schrödinger representation as

$\displaystyle\ket{\psi}=\int_{{\mathbb{R}}^{d}}\differential^{d}x\;\psi(x)\ket{x}_{\mathrm{Sch}}\;,$

(4)

where $\psi\in L^{2}({\mathbb{R}}^{d})$ is the Schrödinger wave function. The momentum operators act on the wave functions as $\hat{p}_{a}\psi(x)\sim-i\hbar\,\frac{\partial}{\partial x^{a}}\psi(x)$.

One can similarly construct the momentum representation from the commutative subalgebra of $\mathcal{W}$ spanned by $\{{\hat{W}_{(a,0)}}\,|\,a\in{\mathbb{R}}^{d}\}$. The momentum eigenvectors $\ket{{\tilde{x}}}_{\mathrm{mom}}$, ${\tilde{x}}\in{\mathbb{R}}^{d}$, satisfy $\hat{p}_{a}\ket{{\tilde{x}}}_{\mathrm{mom}}={\tilde{x}}_{a}\ket{{\tilde{x}}}_{\mathrm{mom}}$, and they are related to the position eigenvectors by a Fourier transform, ${}^{\phantom{*}}_{\mathrm{Sch}}\!\innerproduct{x}{{\tilde{x}}}_{\mathrm{mom}}=\left(2\pi\hbar\right)^{-d/2}e^{ix\cdot{\tilde{x}}/\hbar}$.

After having considered two standard examples, we may now look for the generic commutative subalgebras of the Weyl algebra $\mathcal{W}$. The commutator of two Weyl operators can be written as

$\displaystyle\left[{\hat{W}_{(a,b)}},{\hat{W}_{(a^{\prime},b^{\prime})}}\right]$

$\displaystyle=\left(e^{i(b\cdot a^{\prime}-a\cdot b^{\prime})/\hbar}-1\right)e^{-{\frac{1}{2}}i(b\cdot a^{\prime}-a\cdot b^{\prime})/\hbar}\,{\hat{W}_{(a+a^{\prime},b+b^{\prime})}}\;.$

(5)

This implies that

$\displaystyle\left[{\hat{W}_{(a,b)}},{\hat{W}_{(a^{\prime},b^{\prime})}}\right]=0\qquad\Leftrightarrow\qquad\frac{1}{2\pi\hbar}\left(a^{\prime}\cdot b-a\cdot b^{\prime}\right)\in{\mathbb{Z}}\;.$

(6)

Since this relation is bilinear, the arguments $(a,b)\in{\mathbb{R}}^{2d}$ of the Weyl operators in a generic commutative subalgebra of $\mathcal{W}$ are supported on a lattice in the phase space.

Hereafter, we follow the notation in [2], where double-stroke, capital letters such as ${\mathbb{X}}^{A}=(x^{a},{\tilde{x}}_{a})$ denote a pair of position and momentum variables, which are represented by lowercase letters without and with a tilde, respectively. These composite objects are vectors on the phase space ${\mathcal{P}}={\mathbb{R}}^{d}\oplus{\mathbb{R}}^{d}$.

We introduce a symplectic structure $\omega$ on ${\mathcal{P}}$ by $\omega({\mathbb{X}},{\mathbb{Y}})=\omega_{AB}\,{\mathbb{X}}^{A}\,{\mathbb{Y}}^{B}\equiv{\tilde{x}}\cdot y-x\cdot{\tilde{y}}$ for any ${\mathbb{X}},{\mathbb{Y}}\in{\mathcal{P}}$. We say that $\Lambda\subset{\mathcal{P}}$ is a modular lattice if it is a maximal subset that satisfies $\omega(\Lambda,\Lambda)\subseteq 2\pi\hbar\,{\mathbb{Z}}$. Our discussion above shows that each maximal commutative *-subalgebra ${\mathcal{W}_{\Lambda}}\subset{\mathcal{W}}$ corresponds to a modular lattice $\Lambda$, i.e. it is generated by the elements $\{{\hat{W}_{{\mathbb{K}}}}\,|\,{\mathbb{K}}\in\Lambda\}$.¹¹1The Schrödinger and momentum representations correspond to two singular limits of modular lattices, which we will discuss later in Section 2.4.

In the following, we will assume that the modular lattice $\Lambda$ is of the form $\Lambda=\{(\lambda n,{\tilde{\lambda}}{\tilde{n}})\in{\mathcal{P}}\,|\,n,{\tilde{n}}\in{\mathbb{Z}}^{d}\}$, where $\lambda^{a}{}_{b}$ and ${\tilde{\lambda}}_{a}{}^{b}$ are diagonal $d\times d$-matrices, which satisfy $\lambda^{c}{}_{a}{\tilde{\lambda}}_{c}{}^{b}=2\pi\hbar\,\delta_{a}^{b}$. Note that any modular lattice can be brought to this form by a symplectic coordinate transformation.

The common eigenvectors of ${\mathcal{W}_{\Lambda}}$ are called the modular vectors. A modular vector $\ket{{\mathbb{X}}}_{\Lambda}$, ${\mathbb{X}}\in{\mathcal{P}}$, can be expressed in terms of Schrödinger’s position eigenvectors through a Zak transform [14], such that²²2These modular vectors can equivalently be expressed in terms of momentum eigenvectors as $\displaystyle\ket{{\mathbb{X}}}_{\Lambda}$ $\displaystyle\equiv(\det\lambda)^{-1/2}\,e^{-{\frac{1}{2}}ix\cdot{\tilde{x}}/\hbar}\sum_{{\tilde{n}}\in{\mathbb{Z}}^{d}}e^{-ix\cdot{\tilde{\lambda}}{\tilde{n}}/\hbar}\,|{\tilde{x}}+{\tilde{\lambda}}{\tilde{n}}\rangle_{\mathrm{mom}}^{\phantom{\dagger}}\;.$ (7)

$\displaystyle\ket{{\mathbb{X}}}_{\Lambda}$

$\displaystyle\equiv\big{(}\!\det{\tilde{\lambda}}\big{)}^{-1/2}\,e^{{\frac{1}{2}}ix\cdot{\tilde{x}}/\hbar}\sum_{n\in{\mathbb{Z}}^{d}}e^{i{\tilde{x}}\cdot\lambda n/\hbar}\ket{x+\lambda n}_{\mathrm{Sch}}\;.$

(8)

These vectors satisfy the eigenvalue equation

$\displaystyle{\hat{W}_{{\mathbb{K}}}}\ket{{\mathbb{X}}}_{\Lambda}$

$\displaystyle=e^{{\frac{1}{2}}ik\cdot{\tilde{k}}/\hbar}\,e^{i\omega({\mathbb{K}},{\mathbb{X}})/\hbar}\ket{{\mathbb{X}}}_{\Lambda}\;,\qquad{\mathbb{K}}\in\Lambda\;,\;{\mathbb{X}}\in{\mathcal{P}}\;.$

(9)

for any The action of a generic Weyl operator on a modular vector is given by

$\displaystyle{\hat{W}_{{\mathbb{Y}}}}\ket{{\mathbb{X}}}_{\Lambda}$

$\displaystyle=e^{{\frac{1}{2}}i\omega({\mathbb{Y}},{\mathbb{X}})/\hbar}\ket{{\mathbb{X}}+{\mathbb{Y}}}_{\Lambda}\;,\qquad{\mathbb{X}},{\mathbb{Y}}\in{\mathcal{P}}\;.$

(10)

Moreover, the modular vectors are quasi-periodic under discrete translations along the modular lattice, such that

$\displaystyle\ket{{\mathbb{X}}+{\mathbb{K}}}_{\Lambda}$

$\displaystyle=e^{{\frac{1}{2}}ik\cdot{\tilde{k}}/\hbar}\,e^{{\frac{1}{2}}i\omega({\mathbb{K}},{\mathbb{X}})/\hbar}\ket{{\mathbb{X}}}_{\Lambda}\;,\qquad{\mathbb{K}}\in\Lambda\;,\;{\mathbb{X}}\in{\mathcal{P}}\;.$

(11)

In the following, we will often drop the subscript $\Lambda$ on modular vectors for better readability.

The quasi-periodicity implies that not all modular vectors are linearly independent. In order to construct a basis from the modular vectors, we consider the quotient $T_{\Lambda}\equiv{\mathcal{P}}/\Lambda$, which is called a modular space. Each element³³3We abuse the notation by using the same symbol ${\mathbb{X}}$ both for the elements of ${\mathcal{P}}$ as well as for the corresponding equivalence classes on ${T_{\Lambda}}$. ${\mathbb{X}}\in{T_{\Lambda}}$ of the modular space is an equivalence class of the points $({\mathbb{X}}+\Lambda)\subset{\mathcal{P}}$, which can be represented by any of those points. The modular space is topologically a torus in $2d$ dimensions and it has the volume $(2\pi\hbar)^{d}$. It is the associated Gelfand-Naimark space for a modular representation, which we will regard as a quantum configuration space.

We define a modular cell ${M_{\Lambda}}\subset{\mathcal{P}}$ as any set of representatives of the modular space ${T_{\Lambda}}$. Then, the vectors $\{\ket{{\mathbb{X}}}_{\Lambda}\!\,|\,{\mathbb{X}}\in{M_{\Lambda}}\}$ form a complete and orthonormal basis of the Hilbert space. The orthogonality relation reads

$\displaystyle\innerproduct{{\mathbb{X}}}{{\mathbb{Y}}}=\delta^{2d}({\mathbb{X}}-{\mathbb{Y}})\;,\qquad{\mathbb{X}},{\mathbb{Y}}\in{M_{\Lambda}}\;,$

(12)

where $\delta^{2d}$ denotes the $2d$-dimensional Dirac delta distribution. While the relation (12) gives the inner product of two modular vectors from the same modular cell, the inner product of two generic modular vectors is given by

$\displaystyle\innerproduct{{\mathbb{X}}}{{\mathbb{Y}}}=\sum_{{\mathbb{K}}\in\Lambda}e^{{\frac{1}{2}}ik\cdot{\tilde{k}}/\hbar}\,e^{{\frac{1}{2}}i\omega({\mathbb{K}},{\mathbb{X}})/\hbar}\,\delta^{2d}({\mathbb{X}}-{\mathbb{Y}}+{\mathbb{K}})\;,\qquad{\mathbb{X}},{\mathbb{Y}}\in{\mathcal{P}}\;.$

(13)

The completeness relation for modular vectors reads

$\displaystyle\mathbbm{1}=\int_{{T_{\Lambda}}}\differential^{2d}{\mathbb{X}}\,\outerproduct{{\mathbb{X}}}{{\mathbb{X}}}\;.$

(14)

Note that writing the integration in (14) over the modular space ${T_{\Lambda}}$ employs the fact that $\outerproduct{{\mathbb{X}}}{{\mathbb{X}}}$ is periodic and therefore independent of the choice of the modular cell.

We can write a general quantum state in the modular basis as

$\displaystyle\ket{\phi}=\int_{{T_{\Lambda}}}\differential^{2d}{\mathbb{X}}\;\phi({\mathbb{X}})\ket{{\mathbb{X}}}_{\Lambda}\;,$

(15)

where $\phi({\mathbb{X}})$ is called a modular wave function⁴⁴4This function can be thought of as mapping $\phi:{\mathcal{P}}\rightarrow\mathbb{C}$ under the restriction (16), while a more rigorous definition is given below in terms of $E_{\Lambda}$.. This integral is well-defined only when the integrand $\phi({\mathbb{X}})\ket{{\mathbb{X}}}_{\Lambda}$ is periodic on $T_{\Lambda}$. Therefore, we require the modular wave functions to be also quasi-periodic, such that

$\displaystyle\phi({\mathbb{X}}+{\mathbb{K}})$

$\displaystyle=e^{-{\frac{1}{2}}ik\cdot{\tilde{k}}/\hbar}\,e^{-{\frac{1}{2}}i\omega({\mathbb{K}},{\mathbb{X}})/\hbar}\,\phi({\mathbb{X}})\;,\qquad{\mathbb{K}}\in\Lambda\;,\;{\mathbb{X}}\in{\mathcal{P}}\;.$

(16)

In order to reformulate this statement in a more abstract way as in [2], one may define a $U(1)$-bundle $E_{\Lambda}\rightarrow{T_{\Lambda}}$ over the modular space together with the identification

$\displaystyle E_{\Lambda}:\quad\left(\theta,{\mathbb{X}}\right)\sim\left(\theta\,e^{{\frac{1}{2}}ik\cdot{\tilde{k}}/\hbar}\,e^{{\frac{1}{2}}i\omega({\mathbb{K}},{\mathbb{X}})/\hbar},{\mathbb{X}}+{\mathbb{K}}\right)\;,\quad{\mathbb{K}}\in\Lambda\;,\;{\mathbb{X}}\in{\mathcal{P}}\;,\;\theta\in U(1)\;.$

(17)

Then, the modular wave functions $\phi\in L^{2}(E_{\Lambda})$ correspond to the square-integrable sections of $E_{\Lambda}$.

Finally, we examine the action of Heisenberg operators $\hat{q}^{a}$ and $\hat{p}_{a}$ on a quantum state in the modular representation. After some calculation, we find

	$\displaystyle\hat{q}^{a}\ket{\phi}$	$\displaystyle=\int_{{T_{\Lambda}}}\differential^{2d}{\mathbb{X}}\left(i\hbar\,\frac{\partial}{\partial{\tilde{x}}_{a}}\,\phi({\mathbb{X}})+{\frac{1}{2}}\,x^{a}\,\phi({\mathbb{X}})\right)\ket{{\mathbb{X}}}_{\Lambda}\;,$		(18a)
	$\displaystyle\hat{p}_{a}\ket{\phi}$	$\displaystyle=\int_{{T_{\Lambda}}}\differential^{2d}{\mathbb{X}}\left(-i\hbar\,\frac{\partial}{\partial x^{a}}\,\phi({\mathbb{X}})+{\frac{1}{2}}\,{\tilde{x}}_{a}\,\phi({\mathbb{X}})\right)\ket{{\mathbb{X}}}_{\Lambda}\;.$		(18b)

These equations can be expressed more compactly in terms of an Abelian connection⁵⁵5Unlike all other double-stroke letters in this paper, ${\mathbb{A}}$ denotes a co-vector on ${\mathcal{P}}$, rather than a vector. ${\mathbb{A}}={\mathbb{A}}_{A}\,\differential{\mathbb{X}}^{A}$ on $E_{\Lambda}$, given by

$\displaystyle{\mathbb{A}}_{A}({\mathbb{X}})\equiv\left({\frac{1}{2}}\,{\tilde{x}}_{a},-{\frac{1}{2}}\,x^{a}\right)\;.$

(19)

The key property of this modular connection ${\mathbb{A}}$ is that its curvature form coincides with the symplectic form, i.e. $\differential{\mathbb{A}}=\omega$. Using the modular connection, we can define a covariant derivative $\nabla$, which acts on the modular wave functions as

$\displaystyle\nabla_{A}\phi({\mathbb{X}})$

$\displaystyle\equiv\partial_{A}\phi({\mathbb{X}})+\frac{i}{\hbar}\,{\mathbb{A}}_{A}({\mathbb{X}})\,\phi({\mathbb{X}})\;,$

(20)

where $\partial_{A}\equiv\big{(}\frac{\partial}{\partial x^{a}},\frac{\partial}{\partial{\tilde{x}}_{a}}\big{)}$. Defining also ${\hat{\mathbb{Q}}}^{A}\equiv(\hat{q}^{a},\hat{p}_{a})$, we can finally write the action of Heisenberg operators in the modular representation as ${\hat{\mathbb{Q}}}^{A}\sim i\hbar\,(\omega^{-1})^{AB}\,\nabla_{B}$.

One can check that the actions of the Weyl operators ${\hat{W}_{{\mathbb{Y}}}}$ and the Heisenberg operators ${\hat{\mathbb{Q}}}^{A}$ on a modular wave function preserve the condition (16), therefore these are well-defined operators on the modular Hilbert space $L^{2}(E_{\Lambda})$.

There is a $U(1)$-gauge freedom in defining the modular vectors as follows. For any real, smooth function $\alpha\in C^{\infty}({\mathcal{P}})$ on the phase space, we may redefine the modular vectors as

$\displaystyle\ket{{\mathbb{X}}}_{\Lambda}\rightarrow\ket{{\mathbb{X}}}_{\Lambda}^{\alpha}\equiv e^{i\alpha({\mathbb{X}})}\ket{{\mathbb{X}}}_{\Lambda}\;.$

(21)

While the eigenvalue equation (9) is unaffected by this gauge transformation, the action (10) of a generic Weyl operator on a modular vector becomes

$\displaystyle{\hat{W}_{{\mathbb{Y}}}}\ket{{\mathbb{X}}}_{\Lambda}^{\alpha}$

$\displaystyle=e^{i\alpha({\mathbb{X}})-i\alpha({\mathbb{X}}+{\mathbb{Y}})}\,e^{{\frac{1}{2}}i\omega({\mathbb{Y}},{\mathbb{X}})/\hbar}\ket{{\mathbb{X}}+{\mathbb{Y}}}_{\Lambda}^{\alpha}\;,\qquad{\mathbb{X}},{\mathbb{Y}}\in{\mathcal{P}}\;.$

(22)

Similarly, the gauge transformation changes the quasi-periodicity relation (11) to

$\displaystyle\ket{{\mathbb{X}}+{\mathbb{K}}}_{\Lambda}^{\alpha}$

$\displaystyle=e^{i\beta_{\alpha}({\mathbb{X}},{\mathbb{K}})}\ket{{\mathbb{X}}}_{\Lambda}^{\alpha}\;,\qquad{\mathbb{K}}\in\Lambda\;,\;{\mathbb{X}}\in{\mathcal{P}}\;,$

(23)

where

$\displaystyle\beta_{\alpha}({\mathbb{X}},{\mathbb{K}})$

$\displaystyle\equiv\alpha({\mathbb{X}}+{\mathbb{K}})-\alpha({\mathbb{X}})+\frac{1}{2\hbar}\,k\cdot{\tilde{k}}+\frac{1}{2\hbar}\,\omega({\mathbb{K}},{\mathbb{X}})\;.$

(24)

Hence, it changes the condition (16) accordingly. The $U(1)$-bundle is modified to $E_{\Lambda}^{\alpha}\rightarrow{T_{\Lambda}}$ defined by the identification

$\displaystyle E_{\Lambda}^{\alpha}:\quad\left(\theta,{\mathbb{X}}\right)\sim\left(\theta\,e^{i\beta_{\alpha}({\mathbb{X}},{\mathbb{K}})},{\mathbb{X}}+{\mathbb{K}}\right)\;,\quad{\mathbb{K}}\in\Lambda\;,\;{\mathbb{X}}\in{\mathcal{P}}\;,\;\theta\in U(1)\;.$

(25)

The modular connection ${\mathbb{A}}$ also transforms under this gauge transformation such that

$\displaystyle{\mathbb{A}}_{A}({\mathbb{X}})\rightarrow{\mathbb{A}}_{A}({\mathbb{X}})+\hbar\,\partial_{A}\alpha({\mathbb{X}})\;.$

(26)

Note that the curvature $\omega=\differential{\mathbb{A}}$ of the modular connection is invariant under the gauge transformations.

For modular vectors $\ket{{\mathbb{X}}}_{\Lambda}^{\alpha}$ in a generic gauge $\alpha$, we can write the components of the modular connection as

$\displaystyle{\mathbb{A}}_{A}({\mathbb{X}})={\frac{1}{2}}\,{\mathbb{X}}^{B}\,\omega_{BA}+\hbar\,\partial_{A}\alpha({\mathbb{X}})\;.$

(27)

While the modular vectors defined in the last section had their gauge fixed as $\alpha=0$, we will consider an arbitrary choice of gauge hereafter, even though we often omit the label $\alpha$ for better readability. We will also find out in the next section that a specific gauge fixing is required to obtain the Schrödinger and momentum representations as singular limits of the modular ones.

Roughly speaking, the Schrödinger and momentum representations correspond to the limits of the family of modular representations when the spacing of the modular lattice goes to infinity and zero. In this section, we will discuss the details of this limiting process.

Consider the 1-parameter family of modular lattices $\Lambda=\ell{\mathbb{Z}}^{d}\oplus\tilde{\ell}{\mathbb{Z}}^{d}$, where $\ell$ and $\tilde{\ell}$ are length and momentum scales such that $\ell\tilde{\ell}=2\pi\hbar$. The modular space ${T_{\Lambda}}$ has the size $\ell^{d}\times\tilde{\ell}^{d}$. Recall also that the Heisenberg operators are represented in the modular representations by

$\displaystyle\left(\hat{q}^{a},\hat{p}_{a}\right)\sim\left({\frac{1}{2}}\,x^{a}-\hbar\,\frac{\partial\alpha({\mathbb{X}})}{\partial{\tilde{x}}_{a}}+i\hbar\,\frac{\partial}{\partial{\tilde{x}}_{a}}\,,\,{\frac{1}{2}}\,{\tilde{x}}_{a}+\hbar\,\frac{\partial\alpha({\mathbb{X}})}{\partial x^{a}}-i\hbar\,\frac{\partial}{\partial x^{a}}\right)\;.$

(28)

Now, let’s consider the limit $\ell\rightarrow\infty$. As the position part of the modular space ${T_{\Lambda}}$ grows to infinite size and becomes decompactified, its momentum part shrinks to a point. This has two consequences for the representation of the Heisenberg operators: Firstly, the term $\partial/\partial{\tilde{x}}_{a}$ drops, since the wave functions cannot depend non-trivially on momentum. Secondly, the terms $\hbar\,\partial\alpha({\mathbb{X}})/\partial{\tilde{x}}_{a}$ and ${\tilde{x}}^{a}/2+\hbar\,\partial\alpha({\mathbb{X}})/\partial x^{a}$ must be independent of momentum, otherwise they would become ill-defined in the limit. This implies that $\alpha$ must be of the form $\alpha({\mathbb{X}})=-\frac{1}{2\hbar}\,x\cdot{\tilde{x}}+f(x)$ for the limit $\ell\rightarrow\infty$ to be well-defined. Comparing the representation of the momentum operator to the one in the Schrödinger representation, we find that the gauge choice

$\displaystyle\alpha_{\mathrm{Sch}}({\mathbb{X}})=-\frac{1}{2\hbar}\,x\cdot{\tilde{x}}+\mathrm{const.}\;,$

(29)

is needed to obtain the Schrödinger representation, in which $(\hat{q}^{a},\hat{p}_{a})\sim(x^{a},-i\hbar\,\frac{\partial}{\partial x^{a}})$. We name (29) the Schrödinger gauge. One can check with this gauge fixing that (22) also resembles the action of Weyl operators on Schrödinger eigenvectors given by ${\hat{W}_{{\mathbb{Y}}}}\ket{x}_{\mathrm{Sch}}=e^{{\frac{1}{2}}iy\cdot{\tilde{y}}/\hbar}\,e^{ix\cdot{\tilde{y}}/\hbar}\ket{x+y}_{\mathrm{Sch}}$.

Our argument for (29) is also supported by the quasi-periodicity phase function $\beta_{\alpha_{\mathrm{Sch}}}({\mathbb{X}},{\mathbb{K}})=-k\cdot{\tilde{x}}/\hbar$. Note that this is independent of the momentum winding number ${\tilde{k}}$ as it should be, since the momentum part of the modular space shrinks to a point and any dependency on the momentum winding number would result in an ill-defined phase. A winding number $k$ in the position directions, on the other hand, becomes irrelevant as the configuration space is decompactified.

In the limit $\ell\rightarrow\infty$, the modular lattice transitions to the momentum space. This transition can be understood in a coarse-graining approximation to the momentum space, although it is in fact a singular transition from a discrete set in $2d$ dimensions to a continuous set in $d$ dimensions. The continuous momentum space is qualified as a modular lattice by definition, since it is a maximal subset $\Lambda\subseteq{\mathcal{P}}$ satisfying $\omega(\Lambda,\Lambda)\subset 2\pi\hbar\,{\mathbb{Z}}$, although in fact $\omega(\Lambda,\Lambda)=\left\{0\right\}$. The modular space ${T_{\Lambda}}$ also changes its topology as it becomes the Schrödinger configuration space.

In order to see how the modular vectors behave in the Schrödinger limit, one can expand them in terms of momentum eigenvectors as in the footnote 2. We find

	$\displaystyle\lim_{\ell\rightarrow\infty}\big{(}\!\det{\tilde{\lambda}}\big{)}^{1/2}\,\ket{{\mathbb{X}}}_{\Lambda}^{\alpha_{\mathrm{Sch}}}$	$\displaystyle=\lim_{\tilde{\ell}\rightarrow 0}\left(2\pi\hbar\right)^{-d/2}\big{(}\!\det{\tilde{\lambda}}\big{)}\sum_{{\tilde{n}}\in{\mathbb{Z}}^{d}}e^{-ix\cdot({\tilde{x}}+{\tilde{\lambda}}{\tilde{n}})/\hbar}\,|{\tilde{x}}+{\tilde{\lambda}}{\tilde{n}}\rangle_{\mathrm{mom}}^{\phantom{\dagger}}$
		$\displaystyle=\left(2\pi\hbar\right)^{-d/2}\int_{{\mathbb{R}}^{d}}\differential^{d}{\tilde{x}}\;e^{-ix\cdot({\tilde{x}}+{\tilde{\lambda}}{\tilde{n}})/\hbar}\,|{\tilde{x}}+{\tilde{\lambda}}{\tilde{n}}\rangle_{\mathrm{mom}}^{\phantom{\dagger}}$
		$\displaystyle=\ket{x}_{\mathrm{Sch}}\;.$		(30)

Hence, up to a normalization factor, the modular vectors converge to the position eigenvectors. This concludes our analysis: Although the limit $\ell\rightarrow\infty$ is a singular one in which the topology of the (modular) configuration space changes, we have enough evidence to identify the Schrödinger representation with this limit of modular representations.

A similar discussion applies to the momentum representation in the limit $\ell\rightarrow 0$. However, this limit requires a different choice of gauge fixing, namely

$\displaystyle\alpha_{\mathrm{mom}}({\mathbb{X}})=+\frac{1}{2\hbar}\,x\cdot{\tilde{x}}+\mathrm{const.}\;.$

(31)

To the best of our knowledge, this is the first study in literature that addresses the role of gauge fixing for the singular limits and the fact that Schrödinger and momentum representations require two different choices of modular gauge.

In this section, we will list some key results from Feynman’s path integral in the Schrödinger representation. These are well-known in the literature, but they will be useful later as a reference when we compare them to our new path integral in the modular representation.

The transition amplitude between two position eigenvectors over a finite time interval $[t_{0},t_{f}]$ can be expressed via the path integral

$\displaystyle{}^{\phantom{\dagger}}_{\mathrm{Sch}}\!\!\!\;\bra{x_{f}}e^{-i\;\!(t_{f}-t_{0})\hat{H}/\hbar}\ket{x_{0}}_{\mathrm{Sch}}$

$\displaystyle=\int_{x(t_{0})=x_{0}}^{x(t_{f})=x_{f}}\mathcal{D}x\,\exp\!\left[\frac{i}{\hbar}\,S_{\mathrm{Sch}}[x]\right]\;.$

(34)

On the right-hand side, the functional integral runs over all paths from $x_{0}$ to $x_{f}$ on the configuration space ${\mathbb{R}}^{d}$. The path measure $\mathcal{D}x$ is defined as

$\displaystyle\mathcal{D}x$

$\displaystyle\equiv\lim_{N\rightarrow\infty}\left(\left(\frac{-i}{2\pi\hbar}\,\frac{mN}{t_{f}-t_{0}}\right)^{d/2}\sqrt{\det g}\right)^{N}\prod_{n=1}^{N-1}\differential^{d}x_{n}\;.$

(35)

The action $S_{\mathrm{Sch}}$ is given by

	$\displaystyle S_{\mathrm{Sch}}[x]$	$\displaystyle=\int_{t_{0}}^{t_{f}}\differential t\,\mathcal{L}_{\mathrm{Sch}}(x(t),\dot{x}(t))\;,$		(36a)
	$\displaystyle\mathcal{L}_{\mathrm{Sch}}(x,\dot{x})$	$\displaystyle={\frac{1}{2}}\,m\;\!g(\dot{x},\dot{x})-{\frac{1}{2}}\,m\;\!\Omega^{2}\;\!g(x,x)\;,$		(36b)

where the dot $\dot{}$ over a variable denotes its time derivative. We can make a Legendre transformation on the Lagrangian $\mathcal{L}_{\mathrm{Sch}}$ to recover the classical Hamiltonian function $\mathcal{H}_{\mathrm{Sch}}$, given by

$\displaystyle\mathcal{H}_{\mathrm{Sch}}(x,{\tilde{x}})$

$\displaystyle=\frac{1}{2m}\,g^{-1}({\tilde{x}},{\tilde{x}})+{\frac{1}{2}}\,m\;\!\Omega^{2}\;\!g(x,x)={\frac{1}{2}}\,\Omega\,G((x,{\tilde{x}}),(x,{\tilde{x}}))\;,$

(37)

where ${\tilde{x}}=\partial\mathcal{L}_{\mathrm{Sch}}/\partial\dot{x}$. Defining ${\mathbb{X}}^{A}\equiv(x^{a},{\tilde{x}}_{a})$, we can write the (classical) Hamilton equations as

$\displaystyle\dot{{\mathbb{X}}}(t)=\Omega\,\omega^{-1}G\,{\mathbb{X}}(t)\;.$

(38)

In this section, we will construct, step by step, a path integral formulation for the transition amplitude ${}^{\alpha}_{\Lambda}\bra{{\mathbb{X}}_{f}}e^{-i\;\!(t_{f}-t_{0})\hat{H}/\hbar}\ket{{\mathbb{X}}_{0}}_{\Lambda}^{\alpha}$ between two modular vectors over a finite time interval $[t_{0},t_{f}]$. We assume here that the gauge $\alpha$ is arbitrary, and that the modular lattice is of the form $\Lambda=\lambda{\mathbb{Z}}^{d}\oplus{\tilde{\lambda}}{\mathbb{Z}}^{d}$, where $\lambda$ and ${\tilde{\lambda}}$ are diagonal $d\times d$-matrices that satisfy $\lambda^{c}{}_{a}{\tilde{\lambda}}_{c}{}^{b}=2\pi\hbar\,\delta_{a}^{b}$, as mentioned previously in Section 2.2. We will often omit the labels $\Lambda$ and $\alpha$ on the modular vectors. The Hamiltonian operator is that of a quantum harmonic oscillator given in (32).