Introduction

In light of the inadequacy of the Galilei symmetry to describe electrodynamics, it was necessary to extend it to the Lorentz symmetry, or more generally, the Poincaré symmetry. Through a process of contraction, it was anticipated that the Poincaré symmetry would reproduce the Galilei symmetry. This contraction is characterized by small characteristic velocities compared to the speed of light, referred to as the \(c\rightarrow \infty\) limit or the Galilei/nonrelativistic limit.

Alternatively, the Poincaré symmetry can be contracted by considering characteristic velocities that are much larger than the speed of light, known as the \(c\rightarrow 0\) limit or the Carroll/ultrarelativistic limit. This distinct type of contraction was discovered by Lévy-Leblond in 1965 and is known as the Carroll symmetry1.

Both the Galilei and the Carroll symmetries are two distinct contractions of the Poincaré symmetry, depending on the limit of the speed of light. In terms of the spacetime coordinates, the limit can be expressed as \(x_i\rightarrow \epsilon \,x_i\), \(t\rightarrow t\), \(\epsilon \rightarrow 0\) for the Galilei case, and \(x_i\rightarrow x_i\), \(t\rightarrow \epsilon \,t\), \(\epsilon \rightarrow 0\) for the Carroll case.

While these two contractions are taken from the same thing, it is worth noting that the Galilei symmetry aligns with empirical evidence observed in the natural world, whereas the Carroll symmetry represents a theoretical framework that has the potential to uncover novel aspects of physics.

The study of Carrollian structures and their application to diverse physical systems is driven by several motivations. Notably, a recent motivation that has stimulated this field of research is the fact that the Carrollian conformal algebra is isomorphic to the BMS algebra in one dimension higher2,3,4 and therefore is relevant for the celestial approach to flat space holography5,6,7,8,9,10,11,12,13,14,15,16,17,18.

Other aspects of Carrollian physics has been studied by different motivations, see e.g.19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67 and references therein. In particular, it could potentially have implications for de Sitter cosmology and the phenomenon of inflation44,61, as well as the Hall effect54, hydrodynamics7,28,29, scalar fields44,45,55,60, fermionic fields56,57,63,67, and supersymmetry23,58,63.

The Carrollian physics could offer intriguing possibilities, particularly in the context of quantum mechanics. Usually, the Schrödinger equation can be obtained from the Klein-Gordon equation by applying a field redefinition and taking the Galilei/nonrelativistic limit \(c\rightarrow \infty\) (see e.g.68 or a brief review in Appendix A of the Supplementary Material file). However, in the Carrollian sector, we find a similar approach proves unsuccessful when starting from the Klein-Gordon equation. Instead, one has to begin with the tachyon Klein-Gordon equation.

Consequently, the main objective of this paper is to identify a Carrollian quantum equation, which we call the “Carroll–Schrödinger equation”, analogous to the Schrödinger equation. This is investigated in Sect. Carroll–Schrödinger equation for the case of two dimensions.

The Schrödinger algebra is a conformal extension of the Galilei algebra with a central charge. The field theories with the Schrödinger symmetries have been studied in69,70,71. This algebra captures the symmetry of the Schrödinger equation.

Similarly, another goal of this work is to construct the so-called “Carroll–Schrödinger algebra” in two dimensions, which is a conformal extension of the Carroll algebra with a central charge. Moreover, we illustrate the existence of an infinite-dimensional extension of this algebra. These developments are detailed in Sect. Carroll–Schrödinger algebra.

Furthermore, as detailed in Sect. Symmetries of the action, we demonstrate that our two-dimensional Carrollian quantum equation exhibits the Carroll–Schrödinger algebra as its underlying symmetry.

Additionally, we delve deeper into the Carroll–Schrödinger field theory in Sect. Further study on Carroll–Schrödinger by applying the method of canonical quantization to the theory and computing the transition amplitude.

Finally, in Sect. Comments on higher dimensions, we discuss higher dimensions and derive an equation, called the “generalized Carroll–Schrödinger equation”, which, in two dimensions, reduces to the Carroll–Schödinger equation.

Therefore, throughout this work, while the “generalized Carroll–Schödinger” refers to an equation/action in higher dimensions, the “Carroll–Schödinger” specifically denotes an equation/action/algebra in two dimensions.

The shorthand notations \(\partial _t:=\partial /\partial t\), \(\partial _x:=\partial /\partial x\), and \(\partial _i:=\partial /\partial x^i\) (with \(i=1,\ldots ,d\)) are used.

Carroll–Schrödinger equation

Let us take into account the Klein-Gordon equation for a complex tachyon field \(\phi\) of mass m in a two-dimensional Minkowski spacetime, with the mostly plus signature for the metric, including the speed of light c and the reduced Planck constant \(\hbar\)

$$\begin{aligned} \left( -\,\frac{1}{c^2}\,\partial _t^2\,+\,\partial _x^2\,+\,\mu ^2 \right) \phi =0\,, \end{aligned}$$
(1)

where \(\mu :={mc}/{\hbar }\). By applying a field redefinition

$$\begin{aligned} \phi =\frac{1}{\sqrt{\mu }}~e^{-\,i\mu x}\,\psi \,, \end{aligned}$$
(2)

the tachyon Klein-Gordon equation (1) reduces to

$$\begin{aligned} \left( -\,\frac{1}{c^2}\,\partial _t^2\,-\,2i\mu \,\partial _x\,+\,\partial _x^2\right) \psi =0\,. \end{aligned}$$
(3)

Using a dimensionless parameter \(\epsilon\), we can first rescale \(\mu \rightarrow \mu /\epsilon ^2\) in (3) and then take the Carroll limit

$$\begin{aligned} x\rightarrow x, \qquad t\rightarrow \epsilon \,t,\qquad \epsilon \rightarrow 0. \end{aligned}$$
(4)

Through this process, we obtain the following equation

$$\begin{aligned} {\left( i\hbar \,c\,\partial _x+\frac{\hbar ^2}{2m c^2}\,\partial _t^2 \right) \psi =0\,.} \end{aligned}$$
(5)

We will refer to this equation as the “Carroll–Schrödinger equation”, because it has a Carroll–Schrödinger symmetry, as will be discussed later.

The operator acting on the field \(\psi\) in (5) is Hermitian with respect to the Hermitian conjugation rules: \((\partial _t)^\dagger \equiv -\,\partial _t\), and \((\partial _x)^\dagger \equiv -\,\partial _x\). This suggests that the Carroll–Schrödinger Eq. (5) can be derived from an action principal, which we call the “Carroll–Schrödinger action”

$$\begin{aligned} {S=\int dt\,dx\,\psi ^\star \left( i\hbar \,c\,\partial _x+\frac{\hbar ^2}{2m c^2}\,\partial _t^2 \right) \psi \,.} \end{aligned}$$
(6)

We note that the introduced \(\mu\) in (1) has the length dimension \([\mu ]_{_L}=-1\). In two spacetime dimensions, the length dimension of a scalar field is \([\phi ]_{_L}=0\). Hence, the coefficient in the field redefinition (2) is chosen to ensure that the length dimension of the Carroll–Schrödinger field \(\psi\) becomes \([\psi ]_{_L}=-1/2\). This choice ensures that the action (6) possesses the expected dimension of \(\hbar\).

It is worth noting that when the above procedure is followed and a real scalar field \(\phi\) is employed, such as \(\phi \propto e^{-\mu x}\,\psi\), one arrives at an equation that the operator acting on the field \(\psi\) is not Hermitian, and thus cannot be derived from an action.

Moreover, if we begin with a complex scalar field equation with a real mass, rather than a complex tachyon field Eq. (1), it would result in an equation that becomes ill-defined in the Carroll limit (4). This highlights the necessity of initiating the analysis with a tachyon field Eq. (1).

We note that it is possible to begin with a field redefinition (2) by considering the opposite sign of \(\mu\). This would lead to the replacement of \(m\rightarrow -\,m\) in the resulting equation and action, presenting another possibility.

In our approach, we first derived the equation of motion and subsequently obtained the action. We notice that by beginning with the action of a complex tachyon field, applying the field redefinition (2), and taking the Carroll limit, we could directly arrive at the action (6).

It is also important to emphasize that taking the limit is a technique to obtain a new physics. Accordingly, after taking the limit (4), the nature of initial field has been changed. Instead of being a tachyon, it now represents a massive Carrollian field.

Carroll–Schrödinger algebra

In this section, we introduce the “Carroll–Schrödinger algebra” in two spacetime dimensions, including both finite-dimensional and infinite-dimensional extensions.

Finite-dimensional extension

The Carroll algebra in \(1+1\) dimensions, involving time translation H, space translation P, and Carroll boost B, is given by \([\,P,\,B\,]=H\). It is convenient to see that this algebra admits two non-trivial central extensions as \([\,H,\,B\,]=c_1\), and \([\,H,\,P\,]=c_2\). By ignoring \(c_2\), and identifying \(c_1\) with mass parameter, \(c_1:=M\), the double centrally extended Carroll algebra reduces to

$$\begin{aligned} {~~~ \begin{aligned}&[\,P,\,B\,]=H, \qquad & [\,H,\,B\,]=M. \end{aligned} ~~~} \end{aligned}$$
(7)

We name this the “Carroll–Bargmann algebra”, denoted by \(\mathfrak {carrb}(1+1)\), drawing an analogy to the Bargmann algebra which is a central extension of the Galilei algebra.

Next, we extend the Carroll–Bargmann algebra (7) by including the conformal generators, namely the dilatation D and the spatial special conformal transformation K. We find such an extended algebra, that is a 2-dimensional conformal Carroll algebra with the central charge M, has the following nonzero brackets

$$\begin{aligned} {~~~ \begin{aligned}&[\,P,\,B\,]=H, \quad & [\,H,\,B\,]={M},\\&[\,P,\,D\,]=2\,P, \quad & [\,H,\,D\,]=H, \\&[\,D,\,B\,]=B, \quad & [\,P,\,K\,]=D, \\& [\,D,\,K\,]=2\,K,\quad & [\,H,\,K\,]=B. \end{aligned} ~~~} \end{aligned}$$
(8)

We will refer to this algebra as the “Carroll–Schrödinger algebra”, denoted by \(\mathfrak {carrsch}(1+1)\), because it is the symmetry group of the Carroll–Schrödinger action (6), as will be shown in the next section.

The generators of the Carroll–Schrödinger algebra (8) can be represented by

$$\begin{aligned}&H=\partial _t\,, \\&P=\partial _x\,, \nonumber\\&{M}=-\,i m\,, \nonumber\\& B=x\,\partial _t-i m\,t\,,\nonumber\\ & D=t\,\partial _t+2\,x\,\partial _x+\omega \,, \nonumber \\& K=x\,t\,\partial _t+x^2\,\partial _x-\tfrac{1}{2}\,i m \,t^2+\omega x\,, \end{aligned}$$
(9)

where \(\omega\) is the dilatation weight.

We note that, as demonstrated in (9), the dilatation generator D scales space and time differently \(x\rightarrow \lambda ^2 x\), \(t\rightarrow \lambda \,t\), indicating a critical exponent of \(z=1/2\). However, the corresponding generator in the Carrollian conformal algebra4 scales space and time in the same way \(x \rightarrow \lambda \,x\), \(t \rightarrow \lambda \,t\), with a critical exponent of \(z = 1\).

Infinite-dimensional extension

The \(z=1/2\) Carroll–Schrödinger algebra (8) can be made isomorphic to the \(z=2\) Schrödinger algebra by replacing the generators P and H with each other. This allows us to simply write an infinite-dimensional extension of the algebra (8).

Accordingly, we can present the infinite-dimensional Carroll–Schrödinger algebra, denoted as \(\mathfrak {\widetilde{carrsch}}(1+1)\), as

$$\begin{aligned} \begin{aligned}&[\,L_n,\,L_m\,] = (n-m)\,L_{n+m}, \\&[\,L_n,\,Y_\ell \,] = \left( \,\tfrac{n}{2}-\ell \,\right) Y_{n+\ell }, \\&[\,M_n,\,L_m\,]=n\,M_{n+m},\\&[\,Y_\ell ,\,Y_{k}\,] = (\ell -k)\,M_{\ell +k},\\&[\,Y_\ell ,\,M_{n}\,]=0=[\,M_n,\,M_m\,], \end{aligned} \end{aligned}$$
(10)

where \(n,m\in \mathbb {Z}\), and \(\ell ,k\in \mathbb {Z}+\tfrac{1}{2}\). The infinite-dimensional generators that fulfill the algebra (10) can be given by

$$\begin{aligned} L_n&=-\,x^{n+1}\,\partial _x-\,\tfrac{1}{2}\,(n+1)\,x^n\,t\,\partial _t \,-\tfrac{1}{4}\,M\,n(n+1)\,t^2\,x^{n-1}-\,\tfrac{1}{2}\,\omega \,(n+1)\,x^n\,, \end{aligned}$$
(11)
$$\begin{aligned} Y_\ell&=-\,x^{\,(\ell +\frac{1}{2})}\,\partial _t-\,M\,(\ell +\tfrac{1}{2})\,t\,x^{(\ell -\frac{1}{2})}\,,\end{aligned}$$
(12)
$$\begin{aligned} M_n&=-\,M\,x^n\,, \end{aligned}$$
(13)

where \(\omega\) is the dilatation weight, and M is an arbitrary constant representing the mass parameter.

It is understood that the finite-dimensional generators are those with \(M_0\), \(Y_{\pm \frac{1}{2}}\), \(L_{\pm 1,0}\). In other words, we can write the generators in (9) in a suggestive form

$$\begin{aligned}&M=-\,im=-\,M_0\,, \nonumber \\&B=x\,\partial _t-im\,t=-\,Y_{\frac{1}{2}}\,, \nonumber \\&H=\partial _t=-\,Y_{-\frac{1}{2}}\,, \nonumber \\&P=\partial _x=-\,L_{_{-1}}\,, \nonumber \\&D=t\,\partial _t+2\,x\,\partial _x+\omega =-\,2L_{_0}\,, \nonumber \\&K=x\,t\,\partial _t+x^2\,\partial _x-\tfrac{1}{2}\,i m \,t^2+\omega x=-\,L_{_1}\,. \end{aligned}$$
(14)

As a result, the finite-dimensional Carroll–Schrödinger algebra (8) would be a sub-algebra of (10), leading to the following hierarchy of Lie subalgebras:

$$\begin{aligned} \mathfrak {carr}(1+1)\subset \mathfrak {carrb}(1+1)\subset \mathfrak {carrsch}(1+1) \subset \mathfrak {\widetilde{carrsch}}(1+1). \end{aligned}$$

We note that, in two spacetime dimensions, the \(z=2\) Schrödinger algebra and the \(z=\frac{1}{2}\) Carroll–Schrödinger algebra indeed share the same infinite-dimensional structure, analogous to that of the Carrollian conformal algebra (CCA) and the Galilean conformal algebra (GCA). We also note that the triplet P, D, K (or \(L_{-1}\), \(L_0\), \(L_1\)), forms an sl(2; R) subalgebra.

Symmetries of the action

This section demonstrates that the Carroll–Schrödinger algebra (8) is the symmetry algebra of the Carroll–Schrödinger action (6), while the infinite-dimensional Carroll–Schrödinger algebra (10) does not. This is what we demonstrate in this section, so readers may proceed to the next section without any loss of generality.

The transformation of the Carroll–Schrödinger field can be given by

$$\begin{aligned} \delta \psi&= \big (~\lambda _{_H}\,H~+~\lambda _{_P}\,P~+~\lambda _{_B}\,B~ +~\lambda _{_M}\,M~+~\lambda _{_D}\,D~+~\lambda _{_K}\,K~\big )\,\psi \,, \end{aligned}$$
(15)

where H, P, B, M, D, K are the generators of the Carroll–Schrödinger algebra (8), represented in (9), and \(\lambda _{_H}\), \(\lambda _{_P}\), \(\lambda _{_B}\), \(\lambda _{_M}\), \(\lambda _{_D}\), \(\lambda _{_K}\) are the corresponding parameters associated with each transformation.

Accordingly, we can demonstrate that the Carroll–Schrödinger action is invariant under the transformation (15), upon the dilatation weight \(\omega =1/2\). This invariance implies that the Carroll–Schrödinger action (6), and consequently the Eq. (5), have the Carroll–Schrödinger algebra (8) as their symmetry algebra.

To illustrate this, assuming \(c=1=\hbar\), let us consider the action (6) as

$$\begin{aligned} S=\int dt\,dx~\psi ^\dagger \,\mathbb {K}\,\psi , \end{aligned}$$
(16)

where \(\mathbb {K}\) is defined as the “Carroll–Schrödinger operator”

$$\begin{aligned} \mathbb {K}:=i\,\partial _x+\frac{1}{2m}\,\partial _t^2. \end{aligned}$$
(17)

The variation of the action (16) gives

$$\begin{aligned} \delta S = \int dt\,dx\,\left( \delta \psi ^\dagger \,\mathbb {K}\,\psi +\psi ^\dagger \,\mathbb {K}\,\delta \psi \right) , \end{aligned}$$
(18)

which requires knowing the Hermitian conjugate of \(\delta \,\psi\); i.e. \(\delta \,\psi ^\dagger\). For this purpose, we first find the Hermitian conjugates of the generators in (9). These can be found by applying the Hermitian conjugation rules

$$\begin{aligned} (\partial _t)^\dagger \equiv -\,\partial _t,\quad \quad (\partial _x)^\dagger \equiv -\,\partial _x,\quad \quad t^\dagger \equiv t,\quad \quad x^\dagger \equiv x \end{aligned}$$
(19)

to the Carroll–Schrödinger generators (9), resulting in

$$\begin{aligned}&H^\dagger =-\,\partial _t\,, \nonumber \\&P^\dagger =-\,\partial _x\,, \nonumber \\&M^\dagger =im\,, \nonumber \\ & B^\dagger =-\,x\,\partial _t+i m\,t\,,\nonumber\\& D^\dagger =-\,t\,\partial _t-2\,x\,\partial _x+\omega -3\,, \nonumber\\ & K^\dagger =-\,x\,t\,\partial _t-x^2\,\partial _x+\tfrac{1}{2}\,i m \,t^2+ x\,(\omega -3)\,. \end{aligned}$$
(20)

As a result, using the latter, we can express the Hermitian conjugation of \(\delta \psi\) as follows

$$\begin{aligned} \delta \psi ^\dagger&=\psi ^\dagger \, \big (~\lambda _{_H}\,H^\dagger ~+~\lambda _{_P}\,P^\dagger ~+~\lambda _{_B}\,B^\dagger ~+~\lambda _{_M}\,M^\dagger ~+~\lambda _{_D}\,D^\dagger ~+~\lambda _{_K}\,K^\dagger ~\big )\,. \end{aligned}$$
(21)

Now, we can demonstrate the invariance of the action (16) under each transformation. For example, under the dilatation D, we have

$$\begin{aligned} \delta _{_D} S&=\int dt\,dx\,\left( \delta _{_D}\psi ^\dagger \,\mathbb {K}\,\psi +\psi ^\dagger \,\mathbb {K}\,\delta _{_D}\psi \right) \nonumber \\&=\int dt\,dx\,\lambda _{_D}\psi ^\dagger \Big [\, D^\dagger \, \mathbb {K}+ \mathbb {K}\, D\,\Big ]\psi \nonumber \\&=\int dt\,dx\,\lambda _{_D}\psi ^\dagger \big (2\,\omega -1\big )\,\mathbb {K}\,\psi \,, \end{aligned}$$
(22)

which vanishes for \(\omega =1/2\). Moreover, under the spatial special conformal transformation K, we find

$$\begin{aligned} \!\!\!\delta _{_K} S&=\int dt\,dx\,\left( \delta _{_K}\psi ^\dagger \,\mathbb {K}\,\psi +\psi ^\dagger \,\mathbb {K}\,\delta _{_K}\psi \right) \nonumber \\&=\int dt\,dx\,\lambda _{_K}\psi ^\dagger \!\left[ x\,(2\,\omega -1\big )\,\mathbb {K}+i\left( \omega -\tfrac{1}{2}\right) \right] \psi \,, \end{aligned}$$
(23)

which again vanishes for \(\omega =1/2\). Similarly, it is straightforward to demonstrate the invariance of the action (16) under other transformations.

Let us now examine the invariance of the action under the transformations associated with the generators of the infinite-dimensional algebra (10). For instance, consider the transformation of the field under the generator \(Y_{\ell }\), given by (11),

$$\begin{aligned} \delta _{_Y}\psi =\lambda _{_Y}\,Y_{\ell }~\psi , \end{aligned}$$
(24)

where \(\lambda _{_Y}\) is the transformation parameter. For simplicity, let us examine the invariance at the level of the equation. By varying the Carroll–Schrödinger Eq. (5), assuming \(c=1=\hbar\), under the transformation (24), \((i\,\partial _x+\tfrac{1}{2m}\,\partial _t^2)\,\delta _{_Y}\psi =0\), we obtain

$$\begin{aligned} \lambda _{_Y}\!\left[ -\,m(\ell -\tfrac{1}{2})(\ell +\tfrac{1}{2})\,t\,x^{(\ell -\frac{3}{2})}\right] \psi \approx 0, \end{aligned}$$
(25)

where the “weak equality” symbol \(\approx\) indicates that the equation of motion is applied to reach (25). This relation demonstrates that the differential equation is not invariant under the transformation (24). However, it could be invariant if we choose \(\ell =\pm \,\frac{1}{2}\), corresponding to the invariance under the finite-dimensional generators, i.e. the boost \(B=-Y_{\frac{1}{2}}\) and the time translation \(H=-Y_{-\frac{1}{2}}\). Similarly, we can illustrate that the differential equation would not be invariant under \(L_n\) and \(M_n\), except for the finite-dimensional generators \(L_{\pm 1,0}\) and \(M_0\).

As a result, it is found that only the finite-dimensional Carroll–Schrödinger algebra (8) is the symmetry algebra of the Carroll–Schrödinger equation/action (5)/(6), while the infinite-dimensional algebra (10) does not.

Further study on Carroll–Schrödinger

It is interesting to study the Carroll–Schrödinger field theory in more detail. In this section, we apply the method of canonical quantization and subsequently use it to compute the transition amplitude.

Canonical quantization

The field quantization requires us to consider it as a quantum system rather than a classical one. Therefore, in two spacetime dimensions, we consider the Lagrangian density of the Carroll–Schrödinger field theory as

$$\begin{aligned} \mathcal {L}=\tfrac{1}{2}\,i\hbar c\,\Big [\psi ^\dagger \partial _x\psi -(\partial _x\psi ^\dagger )\psi \Big ]+\frac{\hbar ^2}{2mc^2}\,\psi ^\dagger \partial _t^2\,\psi , \end{aligned}$$
(26)

where the fields \(\psi\) and \(\psi ^\dagger\) are treated as operators. We note that this form of the Lagrangian density is indeed equivalent to the one in (6) up to a total derivative. The advantage of this form is that we can now introduce canonical momenta conjugate to both fields \(\psi\) and \(\psi ^\dagger\), as derived below. In contrast, using the previous form in (6), only one canonical momentum can be defined.

A general solution of the Carroll–Schrödinger equation for \(\psi\), and its complex conjugate for \(\psi ^\dagger\), can be given by the plane-wave solutions

$$\begin{aligned} \psi (x,t)&=\int \,\frac{dE}{2\pi \hbar c}~a(E)\,e^{-\,i\left( Px-Et\right) /\hbar }\,,\end{aligned}$$
(27)
$$\begin{aligned} \psi ^\dagger (x,t)&=\int \,\frac{dE}{2\pi \hbar c}~a^\dagger (E)\,e^{i\left( Px-Et\right) /\hbar }\,, \end{aligned}$$
(28)

where the coefficients a(E) and \(a^\dagger (E)\) are also operators with the length dimension of 1/2, and

$$\begin{aligned} P=\frac{E^2}{2mc^3} \end{aligned}$$
(29)

is the energy-momentum relation in the Carrollian framework, as detailed in72. We note that negative energies are possible within the Carrollian framework, since it was derived from a tachyonic scenario. This can also be confirmed by (29), where momentum is positive definite, but energy can be negative. This is in contrast to the Galilean case, where energy is positive definite, \(E=P^2/2m\), while momentum can be negative.

We then define new canonical momenta conjugate to the fields \(\psi\) and \(\psi ^\dagger\), subject to their spatial derivatives, as follows (a similar definition was used in63):

$$\begin{aligned} \pi&:=\frac{\partial \mathcal {L}}{\partial \mathring{\psi }}=i\hbar c\,\psi ^\dagger \,, ~~~~~~\qquad \mathring{\psi }\equiv \tfrac{1}{2}\, \partial _x\psi \,,\end{aligned}$$
(30)
$$\begin{aligned} \pi ^\dagger&:=\frac{\partial \mathcal {L}}{\partial \mathring{\psi ^\dagger }}=-\,i\hbar c\,\psi \,, ~~\qquad \mathring{\psi ^\dagger }\equiv \tfrac{1}{2}\, \partial _x\psi ^\dagger \,. \end{aligned}$$
(31)

Accordingly, in the Carrollian framework, we can present the so-called “equal-position commutation relations”

$$\begin{aligned} &[\,\psi(x,t)\,,\,\pi (x,t')\,]=i\hbar \,\delta (t-t'),\nonumber \\ &[\,\psi (x,t)\,,\,\psi (x,t')\,]=[\,\pi (x,t)\,,\,\pi (x,t')\,]=0\,, \end{aligned}$$
(32)

and similar relations for their Hermitian conjugates. By applying (27) and (30) in the quantization conditions (32), we can conveniently find the commutation relations between the operators a and \(a^\dagger\), which become

$$\begin{aligned}&[\,a(E)\,,\,a^\dagger (E')\,]=2\pi \hbar \,c\,\delta (E-E')\,,\nonumber \\&[\,a(E)\,,\,a(E')\,]=[\,a^\dagger (E)\,,\,a^\dagger (E')\,]=0\,. \end{aligned}$$
(33)

These commutation relations can also be satisfied by

$$\begin{aligned} a(E)&=\int c \,dt~\psi (x,t)~e^{i\left( Px-Et\right) /\hbar }\,,\end{aligned}$$
(34)
$$\begin{aligned} a^\dagger (E)&=\int c\, dt~\psi ^\dagger (x,t)~e^{-\,i\left( Px-Et\right) /\hbar }\,. \end{aligned}$$
(35)

When we posit the existence of a vacuum state \(|\,0\,\rangle\), such that \(a(E)|\,0\,\rangle =0\), the particle picture emerges. The operator a(E) annihilates a particle of energy E, while \(a^\dagger (E)\) creates one. Therefore, \(\psi\) annihilates particles and \(\psi ^\dagger\) creates particles.

We can continue our discussion on canonical quantization, but let us delve into more detail in a future work. For now, with this information, we are equipped to compute the transition amplitude in the next section.

Transition amplitude

In two spacetime dimensions, we study the transition amplitude (or two-point function) for the transition of a free particle from a spacetime position \(\textrm{X}\equiv (x^0,x)\) to a position \(\textrm{Y}\equiv (y^0,y)\), with \(y^0>x^0\). The transition amplitude is given by

$$\begin{aligned} \mathbb {M}_{_{\,\textrm{X}\rightarrow \textrm{Y}}}=\langle \,\textrm{Y}\,|\,\textrm{X}\,\rangle , \end{aligned}$$
(36)

where \(|\textrm{X}\rangle\), \(|\textrm{Y}\rangle\) are states of a single particle at positions X, Y correspondingly. These states are given by the action of field on the vacuum

$$\begin{aligned} |\,\textrm{X}\,\rangle =\psi ^\dagger (x^0,x)\,|\,0\,\rangle ,\qquad |\,\textrm{Y}\,\rangle =\psi ^\dagger (y^0,y)\,|\,0\,\rangle . \end{aligned}$$
(37)

The transition amplitude (36) is then

$$\begin{aligned} \mathbb {M}_{_{\,\textrm{X}\rightarrow \textrm{Y}}}&=\langle \,0\,|\, \psi (y^0,y)\,\psi ^\dagger (x^0,x)\, |\,0\,\rangle =\int \,\frac{dE}{2\pi \hbar \,c}~e^{-\,i\left( P\Delta _x\,-\,E\Delta _t\right) /\hbar }\,, \end{aligned}$$
(38)

where

$$\begin{aligned} \Delta _t=y^0-x^0,\qquad \Delta _x=y-x. \end{aligned}$$
(39)

To arrive this, we have employed (27), (28), utilized the quantization condition (33), and the fact that \(a|\,0\,\rangle =0\). By substituting (29) into the integral (38) and completing the square, it can be transformed into a Gaussian integral. Upon evaluating this Gaussian integral, the transition amplitude (38) becomes

$$\begin{aligned} {~ \mathbb {M}_{_{\,\textrm{X}\rightarrow \textrm{Y}}}=\sqrt{\frac{mc}{2\pi i\hbar \,\Delta _x}}~\exp \left( {\frac{imc^3}{2\hbar }\,\frac{(\Delta _t)^2}{\Delta _x}}\right) , ~} \end{aligned}$$
(40)

which indicates a localization in space, as expected in the Carrollian framework. This result is consistent with the Schrödinger case, by exchanging \(\Delta _x \leftrightarrow c\,\Delta _t\).

It is worth noting that in this section, we utilized a method to quantize the Carrollian system, allowing us to derive the two-point function through field theory, as shown in (40). This result is consistent with the findings in Ref.73, which employs an alternative approach that does not rely on field theory, validating our quantization method.

Comments on higher dimensions

Up to this point, we have focused on two spacetime dimensions, but it becomes intriguing to explore the possibilities of extending beyond two dimensions. As discussed in previous sections, this can be approached both at the level of theory and at the level of algebra.

At the level of theory: We can proceed further since the tachyon Klein-Gordon equation (1) can be considered in any dimension. Therefore, we begin with the Klein-Gordon equation applied to a complex scalar tachyon field \(\phi\) in any spatial dimension

$$\begin{aligned} \left( -\,\frac{1}{c^2}\,\partial _t^2\,+\,\partial ^i\partial _i\,+\,\mu ^2\right) \phi =0. \end{aligned}$$
(41)

By applying the field redefinition

$$\begin{aligned} \phi =\frac{1}{\sqrt{\mu }}~e^{-\,i\mu \sqrt{x^2}}\,\psi , \end{aligned}$$
(42)

with \(x^2=x^ix_i\), we can arrive at the equation

$$\begin{aligned} \hspace{-11pt}\left( -\,\frac{1}{c^2}\,\partial _t^2-2i\mu \,\frac{1}{\sqrt{x^2}}\,x^j\partial _j-i\mu \,\frac{d-1}{\sqrt{x^2}}+\partial ^j\partial _j\right) \psi =0. \end{aligned}$$
(43)

After rescaling the parameter \(\mu \rightarrow \mu /\epsilon ^2\) in the latter and then applying the Carroll limit (4), we reach to

$$\begin{aligned} {\begin{aligned}&\left( i\hbar c\,\nabla _x+\frac{\hbar ^2}{2m c^2}\,\partial _t^2\right) \psi =0, \\&\text {where}~~~~~~~\nabla _x=\frac{1}{\sqrt{x^2}}\left( x^i\partial _i+\frac{d-1}{2}\right) . \end{aligned}} \end{aligned}$$
(44)

When \(d=1\), the operator \(\nabla _x\) simplifies to the standard derivative \(\nabla _x \rightarrow \partial _x\), and thus (44) reduces to the Carroll–Schrödinger equation (5). Despite this equality in two dimensions and the fact that we have applied the Carroll limit (4) to derive the Eq.(44) in higher dimensions, we refrain from naming it the “Carroll–Schrödinger equation in higher dimensions”. Instead, we refer to (44) as the “generalized Carroll–Schrödinger equation”.

We observe that the operator \(\nabla _x\) has a length dimension of \([\nabla _x]_{_L} = -1\). Moreover, it is anti-Hermitian, i.e. \((\nabla _x)^{\,\dagger } = -\nabla _x\), with respect to the Hermitian conjugation already introduced. This guarantees that in (44), the operator acting on the field \(\psi\) is Hermitian. Consequently, the equation (44) can be derived from an action, which we call the “generalized Carroll–Schrödinger action”

$$\begin{aligned} {S=\int dt\,d^dx\,\psi ^\star \left( i\hbar c\,\nabla _x+\frac{\hbar ^2}{2m c^2}\,\partial _t^2\right) \psi .} \end{aligned}$$
(45)

Recalling once again that in d spatial dimension, the length dimension of a scalar field is \([\phi ]_{_L}=(1-d)/2\), we can see from (42) that the length dimension of the generalized Carroll–Schrödinger field is \([\psi ]_{_L}=-\,d/2\). This satisfies the expected dimension of \(\hbar\) for the action (45) and is the same as that of the Schrödinger field in any dimension (see appendix A).

Once again, in our approach, we first derived the equation of motion (44) and subsequently obtained the action. We notice that by beginning with the action of a complex tachyon field, applying the field redefinition (42), and taking the Carroll limit, we could directly arrive at the action (45).

The Schrödinger equation does not hold invariance under the transformation \(t\rightarrow -\,t\). However, in the case of the Eq. (44), one finds that it is invariant under both the transformations \(t\rightarrow -\,t\) and \(x_i \rightarrow -\,x_i\) for \(d>1\), while it is not invariant for \(d=1\). This is similar to the ultrarelativistic wave equation21, \((-\,\partial _t^2-m^2)\phi =0\), which also remains invariant under these transformations in any dimension.

We note that the Carroll–Schrödinger equation (5) is mathematically identical to the Schrödinger equation in two spacetime dimensions when the time and spatial coordinates are exchanged (\(x\leftrightarrow c\,t\)). However, it is important to note that this identification only holds in two dimensions and breaks down in higher dimensions, as can be seen by comparing the generalized Carroll–Schrödinger equation (44) and the d-dimensional Schrödinger equation.

At the level of algebra: To determine the symmetry algebra of the generalized Carroll–Schrödinger equation (44), one approach is to extend the Carroll–Schrödinger algebra (8) to higher dimensions. However, we find that the Jacobi identity can only be satisfied in two dimensions. This aligns with the results in73, which derived the Carroll–Schrödinger algebra only in two dimensions using a different method. Consequently, the symmetry algebra of the generalized Carroll–Schrödinger equation (44) remains unknown. Extending the algebra (8) to higher dimensions may require activating some of the previously vanished commutators, a task we plan to explore in the future.

Discussion

In the specific case of two spacetime dimensions, we have formulated a novel equation, which we have named the Carroll–Schrödinger equation (5). Our derivation initially stemmed from a relativistic tachyon equation; however, it is crucial to highlight that the resulting equation does not exhibit tachyonic behavior. Instead, it embodies the nature of Carrollian dynamics.

In addition, we have successfully achieved a conformal extension of the centrally extended Carroll algebra in two dimensions. This extension, called the Carroll–Schrödinger algebra (8), could be extended to the infinite-dimensional version (10) demonstrating a same structure as that of the Schrödinger algebra. We found that the Carroll–Schrödinger algebra (8) serves as the symmetry algebra for the derived action (6), while the infinite-dimensional version does not.

Moreover, we further explored the Carroll–Schrödinger field theory by applying the method of canonical quantization and computing the transition amplitude (40). The latter corresponded with the results in73.

Furthermore, we attempted to extend our results to higher dimensions. By considering our initial framework in higher dimensions and applying the Carroll limit, we derived an equation in higher dimensions, referred to as the generalized Carroll–Schrödinger equation (44). In two dimensions, this equation appropriately reduces to the Carroll–Schrödinger equation (5).

It becomes an intriguing question to explore the possibilities of extending the Carroll–Schrödinger algebra beyond two dimensions. In arbitrary dimensions, we found that the Jacobi identity cannot be satisfied, except in two dimensions. This suggests that extending the algebra (8) to higher dimensions may require activating some of the previously vanished commutators. Such an extension to higher dimensions may serve as the symmetry algebra for the generalized Carroll–Schrödinger equation (44), a task we plan to explore in the future.

The action (45) demonstrates a global phase symmetry under the transformation \(\psi \rightarrow e^{i\lambda }\,\psi\). Therefore, exploring a local phase symmetry could present an interesting research problem.

Moreover, as we know, the Schrödinger equation can also be derived from the Dirac equation (see e.g.68). Consequently, it becomes interesting to explore the possibility of deriving the Carroll–Schrödinger equation (5) from a tachyonic Dirac equation.

In both Galilean and Carrollian field theories, it is known that there are two distinct sectors: the electric sector and the magnetic sector. A comprehensive review of electric and magnetic scalar fields, as well as electric and magnetic spinor fields, can be found in63. It is noteworthy that both the electric and magnetic versions share the same underlying symmetry algebra, whether it be Galilean or Carrollian.

With this in mind, if we refer to the Schrödinger equation as the electric version, it raises the question of what the magnetic version of the Schrödinger equation would be. Similarly, if we consider the obtained Carroll–Schrödinger equation (5) as the magnetic version, we may inquire about its electric counterpart. One possible approach to deriving such equations is through the utilization of the seed Lagrangian method, presented in55 (for further details, refer also to63).

The Carroll–Schrödinger equation (5) and the generalized equation (44) can provide a promising avenue for the development of Carrollian quantum mechanics, with potential applications in condensed matter systems. This could open up new possibilities for addressing various problems within the Carrollian framework, such as: the infinite potential well, the Aharanov Böhm Effect, the Hydrogen atom, the harmonic oscillator and so on. As we were finalizing this work, we came across this reference74 that derived the two-dimensional equation (5) by a different method. It also solved the problem of the infinite potential well.

By solving the equation for different potential energy scenarios, we can compare the results with established outcomes and examine any differences. Notably, it seems the presence of the speed of light c in the Carroll–Schrödinger equation (5) has posed challenges in finding the effects of Carrollian quantum mechanics so far. However, it is anticipated that a careful analysis may reveal potential shifts in energy levels associated with this novel framework.