Conformal Symmetry Of Free Schrodinger Field Theory

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Introduction

In the realm of theoretical physics, conformal symmetry plays a pivotal role in understanding the behavior of physical systems at critical points and in various high-energy scenarios. This article delves into the intricate details of conformal symmetry within the framework of free Schrödinger field theory. Specifically, we will explore how the action of a free Schrödinger field theory in d dimensions transforms under conformal rescalings. The analysis involves a careful examination of the transformations and their implications on the underlying physics. Understanding conformal symmetry is crucial for grasping the fundamental aspects of quantum field theories and their applications in diverse areas such as condensed matter physics, string theory, and statistical mechanics. The Schrödinger field theory, while non-relativistic, exhibits rich symmetries that are vital for modeling systems with non-relativistic scaling behavior, making it an essential topic for both theoretical investigations and practical applications. This exploration will not only enhance our understanding of the symmetries inherent in the Schrödinger equation but also provide insights into the broader context of conformal field theories and their significance in modern physics.

The study of conformal symmetry in Schrödinger field theory is essential for several reasons. Firstly, it provides a concrete example of a non-relativistic system exhibiting conformal invariance, which is a relatively rare phenomenon. This allows physicists to test and refine their understanding of conformal symmetries in a simpler, more manageable setting compared to relativistic conformal field theories. Secondly, Schrödinger-invariant field theories are directly relevant to various physical systems, including ultracold atoms at unitarity, certain condensed matter systems, and even non-relativistic string theories. By understanding the conformal symmetries of these theories, we can gain deeper insights into the behavior of these systems and make more accurate predictions about their properties. Furthermore, the mathematical techniques developed for analyzing conformal symmetry in Schrödinger field theory can be adapted and applied to other non-relativistic field theories, making it a valuable tool in the broader field of theoretical physics. The exploration of these symmetries not only deepens our theoretical understanding but also opens up new avenues for experimental verification and application in real-world physical systems. The ability to connect theoretical concepts with experimental observations is a hallmark of successful physics, and the study of conformal symmetry in this context is a significant step in that direction.

Lastly, the investigation of conformal symmetry in free Schrödinger field theory serves as a pedagogical stepping stone for understanding more complex conformal field theories (CFTs). The free theory provides a simplified environment where the basic concepts and techniques of conformal symmetry can be introduced and mastered without the added complications of interactions. This makes it an ideal starting point for students and researchers new to the field. By working through the details of the free theory, one can develop a solid foundation for tackling more advanced topics such as interacting CFTs, conformal anomalies, and the conformal bootstrap. The insights gained from studying the free theory also provide valuable intuition for understanding the structure and properties of CFTs in general, which are ubiquitous in various areas of physics, ranging from critical phenomena to string theory. Therefore, the study of conformal symmetry in free Schrödinger field theory is not just an end in itself but also a crucial step in the broader journey of understanding the fundamental principles of theoretical physics.

Conformal Transformations

To understand the variation of the action under conformal rescaling, we must first define the transformations under consideration. In this context, we are examining transformations of time and space coordinates that preserve angles locally. Specifically, the transformations are given by:

t→t′=f(t),xi→xi′=g(t)Rij(t)xjt \rightarrow t' = f(t), \qquad x_i \rightarrow x'_i = g(t) R_{ij}(t) x_j

where f(t) and g(t) are functions of time, and Rᵢⱼ(t) is a time-dependent rotation matrix. These transformations generalize the usual scaling and rotations found in standard conformal transformations. The time transformation f(t) represents a general time rescaling, while g(t) scales the spatial coordinates, and Rᵢⱼ(t) accounts for rotations in the d-dimensional space. These transformations collectively form the Schrödinger group, which is the symmetry group of the free Schrödinger equation. The Schrödinger group includes time translations, spatial translations, Galilean boosts, spatial rotations, dilations (scaling transformations), and special conformal transformations. Understanding how these transformations affect the Schrödinger equation and the corresponding action is crucial for analyzing the conformal symmetry of the system.

The significance of these transformations lies in their ability to leave the form of the Schrödinger equation invariant. This invariance is not as strict as in relativistic conformal field theories, where the metric itself is invariant up to a scale factor. Instead, the Schrödinger invariance is a weaker form of conformal symmetry that preserves the structure of the equation under the specified transformations. This is particularly relevant for systems exhibiting non-relativistic scaling behavior, such as ultracold atoms at unitarity and certain condensed matter systems. The transformations f(t) and g(t) dictate how time and space coordinates are rescaled, while the rotation matrix Rᵢⱼ(t) ensures that the spatial rotations are properly accounted for. The combination of these transformations allows us to explore the symmetry properties of the Schrödinger equation and the associated field theories. By analyzing how the action changes under these transformations, we can identify the conditions under which the theory exhibits conformal invariance and understand the implications for the physical observables.

Moreover, the specific forms of f(t), g(t), and Rᵢⱼ(t) determine the particular conformal transformations under consideration. For example, a simple time translation corresponds to f(t) = t + a, where a is a constant. Dilations, which are scaling transformations, can be represented by f(t) = λ²t and g(t) = λ, where λ is a scaling factor. Special conformal transformations are more complex and involve quadratic terms in time. The rotation matrix Rᵢⱼ(t) allows for time-dependent rotations, which are crucial for preserving the symmetry of the Schrödinger equation in higher dimensions. The interplay between these different types of transformations gives rise to the rich structure of the Schrödinger group. By systematically analyzing the effects of these transformations on the action, we can gain a deeper understanding of the conformal symmetry of the free Schrödinger field theory and its implications for the behavior of the system.

Action of Free Schrödinger Field Theory

The action for a free Schrödinger field theory in d dimensions is given by:

S=∫dtddx  iψ∗∂tψ−12m∇ψ∗⋅∇ψS = \int dt d^dx \; i\psi^* \partial_t \psi - \frac{1}{2m} \nabla \psi^* \cdot \nabla \psi

where ψ(t, x) is the complex scalar field, ψ(t, x)* is its complex conjugate, m is the mass of the field, and ∇ represents the spatial gradient. This action describes the dynamics of a non-relativistic free particle and is fundamental to understanding various physical phenomena, such as the behavior of non-interacting bosons in a trap or the dynamics of particles near a Feshbach resonance. The action is composed of two terms: the first term, iψ ∂ₜ ψ*, represents the kinetic energy of the field, and the second term, (1/2m) ∇ψ · ∇ψ*, corresponds to the potential energy. The interplay between these two terms determines the evolution of the field and its correlation functions. The action is invariant under certain transformations, which correspond to the symmetries of the theory. Identifying these symmetries is crucial for understanding the underlying physics and for constructing more complex interacting theories.

This action is a cornerstone of non-relativistic quantum field theory, and its properties have been extensively studied. The equation of motion derived from this action is the free Schrödinger equation, which describes the time evolution of the field. The solutions to this equation form the basis for understanding the behavior of non-interacting particles. The action is also essential for calculating correlation functions, which provide information about the statistical properties of the field. The invariance of the action under certain transformations, such as time translations, spatial translations, and Galilean boosts, reflects the fundamental symmetries of the system. In addition to these standard symmetries, the action also exhibits conformal symmetry under certain transformations, which is the focus of this article. Understanding the conformal symmetry of the action is crucial for analyzing the behavior of the system under scaling transformations and for constructing conformal field theories.

Moreover, the form of the action dictates the structure of the theory and its interactions. In this case, the action is quadratic in the fields, which means that the theory is free and there are no interactions between the particles. This simplifies the analysis and allows for exact solutions. However, the free theory can also serve as a starting point for studying interacting theories, where additional terms are added to the action to account for the interactions. The symmetries of the free theory, including conformal symmetry, can provide valuable guidance for constructing consistent interacting theories. By understanding how the action transforms under conformal transformations, we can identify the conditions under which the theory remains invariant and construct operators that transform covariantly. This is essential for building a conformal field theory that describes the behavior of the system at a critical point or in a high-energy regime. Therefore, the action of the free Schrödinger field theory is not only a fundamental object in its own right but also a crucial building block for more complex theories.

Variation of the Action

To calculate the variation of the action under the conformal transformations, we need to consider how the fields and the integration measure transform. Under the transformations, the fields transform as:

ψ(t,x)→ψ′(t′,x′)=Ω(t,x)ψ(t,x)\psi(t, x) \rightarrow \psi'(t', x') = \Omega(t, x) \psi(t, x)

ψ∗(t,x)→ψ∗′(t′,x′)=Ω∗(t,x)ψ∗(t,x)\psi^*(t, x) \rightarrow \psi^{*'}(t', x') = \Omega^*(t, x) \psi^*(t, x)

where Ω(t, x) is a spacetime-dependent scaling factor. The scaling factor is crucial for ensuring the invariance or covariance of the action under the transformations. The specific form of Ω(t, x) depends on the conformal transformation being considered and the scaling dimension of the field ψ. In addition to the fields, the integration measure also transforms. The transformation of the integration measure is given by:

dt  ddx→dt′  ddx′=∣J(t,x)∣dt  ddxdt \; d^dx \rightarrow dt' \; d^dx' = |J(t, x)| dt \; d^dx

where J(t, x) is the Jacobian determinant of the transformation. The Jacobian determinant accounts for the change in volume element under the transformation and is essential for ensuring the proper normalization of integrals. The variation of the action is then calculated by substituting these transformations into the action and examining the resulting expression. This involves careful application of the chain rule and integration by parts. The goal is to identify terms that vanish due to the transformations and to determine the conditions under which the action remains invariant or transforms in a covariant manner.

The calculation of the variation of the action is a crucial step in understanding the conformal symmetry of the theory. By explicitly computing how the action changes under the conformal transformations, we can identify the conserved currents and charges associated with the symmetry. These conserved quantities provide valuable information about the dynamics of the system and can be used to constrain the correlation functions. The scaling factor Ω(t, x) plays a critical role in ensuring the invariance or covariance of the action. Its form is determined by the requirement that the transformed action has the same form as the original action, possibly up to a total derivative term. The Jacobian determinant J(t, x) is also essential for maintaining the correct normalization of the action and for ensuring that the equations of motion remain invariant under the transformations.

Furthermore, the variation of the action can reveal the presence of conformal anomalies, which are violations of conformal symmetry at the quantum level. These anomalies arise from the regularization and renormalization procedures required to make the theory well-defined. They manifest themselves as terms in the transformed action that are not present in the original action. The presence of conformal anomalies can have significant implications for the behavior of the theory, particularly at critical points. By carefully calculating the variation of the action and identifying the anomalous terms, we can gain a deeper understanding of the quantum properties of the conformal field theory. This analysis is crucial for constructing consistent and physically meaningful theories. The results of this calculation will ultimately determine whether the action is invariant under the given transformations and what conditions are necessary for conformal invariance.

Detailed Calculation

To proceed with the detailed calculation, let's consider the specific transformations:

t→t′=f(t)t \rightarrow t' = f(t)

xi→xi′=g(t)xix_i \rightarrow x'_i = g(t) x_i

For simplicity, we will assume that the rotation Rᵢⱼ(t) is the identity matrix. The Jacobian determinant for this transformation is:

J(t,x)=∂t′∂t∏i=1d∂xi′∂xi=f′(t)g(t)dJ(t, x) = \frac{\partial t'}{\partial t} \prod_{i=1}^d \frac{\partial x'_i}{\partial x_i} = f'(t) g(t)^d

where f'(t) denotes the derivative of f(t) with respect to t. The transformation of the fields is given by:

ψ′(t′,x′)=Ω(t,x)ψ(t,x)\psi'(t', x') = \Omega(t, x) \psi(t, x)

ψ∗′(t′,x′)=Ω∗(t,x)ψ∗(t,x)\psi^{*'}(t', x') = \Omega^*(t, x) \psi^*(t, x)

The scaling dimension Δ of the field ψ determines the form of the scaling factor Ω(t, x). For a free Schrödinger field, the scaling dimension is typically given by Δ = d/2. The scaling factor is then:

Ω(t,x)=g(t)−d/2\Omega(t, x) = g(t)^{-d/2}

The derivatives transform as:

∂t=∂t′∂t∂t′=f′(t)∂t′\partial_t = \frac{\partial t'}{\partial t} \partial_{t'} = f'(t) \partial_{t'}

∂xi=∂xj′∂xi∂xj′=g(t)∂xi′\partial_{x_i} = \frac{\partial x'_j}{\partial x_i} \partial_{x'_j} = g(t) \partial_{x'_i}

Substituting these transformations into the action, we get:

S′=∫dt′ddx′  iψ∗′∂t′ψ′−12m∇′ψ∗′⋅∇′ψ′S' = \int dt' d^dx' \; i\psi^{*'} \partial_{t'} \psi' - \frac{1}{2m} \nabla' \psi^{*'} \cdot \nabla' \psi'

S′=∫dt  ddx  f′(t)g(t)d[iΩ∗ψ∗1f′(t)∂t(Ωψ)−12mg(t)−2Ω∗∇ψ∗⋅∇(Ωψ)]S' = \int dt \; d^dx \; f'(t) g(t)^d \left[ i \Omega^* \psi^* \frac{1}{f'(t)} \partial_t (\Omega \psi) - \frac{1}{2m} g(t)^{-2} \Omega^* \nabla \psi^* \cdot \nabla (\Omega \psi) \right]

Expanding the derivatives and simplifying, we can identify the conditions under which the action remains invariant or transforms covariantly. This involves careful analysis of the terms and their dependence on f(t) and g(t). The goal is to determine the constraints on f(t) and g(t) that ensure the conformal invariance of the action.

This detailed calculation is essential for understanding the behavior of the action under conformal transformations. By explicitly substituting the transformations and simplifying the expression, we can identify the terms that contribute to the variation of the action. The scaling dimension of the field plays a crucial role in determining the form of the scaling factor Ω(t, x), which in turn affects the transformation of the action. The derivatives transform according to the chain rule, and their transformation properties must be carefully taken into account. The Jacobian determinant ensures that the integration measure is properly transformed, which is essential for maintaining the correct normalization of the action.

Furthermore, the expansion of the derivatives and the subsequent simplification involve several steps that require careful attention to detail. The goal is to isolate the terms that depend on the derivatives of f(t) and g(t), as these terms will determine the conditions for conformal invariance. The action remains invariant under the transformations if the terms involving the derivatives of f(t) and g(t) vanish. Alternatively, the action transforms covariantly if these terms can be expressed as a total derivative, which does not affect the equations of motion. By carefully analyzing the resulting expression, we can identify the specific transformations that preserve the conformal symmetry of the free Schrödinger field theory. This analysis provides valuable insights into the structure and properties of the theory and its applications in various physical systems.

Results and Conclusion

After performing the detailed calculation and simplifying the expression for the transformed action, we can analyze the results to determine the conditions for conformal invariance. The variation of the action depends on the specific forms of f(t) and g(t), which dictate the conformal transformation under consideration. For the action to be invariant under the transformations, the terms involving the derivatives of f(t) and g(t) must vanish or be expressible as a total derivative. This leads to specific constraints on the functions f(t) and g(t), which define the allowed conformal transformations.

The analysis reveals that the free Schrödinger field theory exhibits conformal symmetry under a specific set of transformations, which form the Schrödinger group. These transformations include time translations, spatial translations, Galilean boosts, spatial rotations, dilations, and special conformal transformations. The invariance of the action under these transformations implies the existence of conserved currents and charges, which are associated with the symmetries. These conserved quantities provide valuable information about the dynamics of the system and can be used to constrain the correlation functions.

In conclusion, the calculation of the variation of the action of free Schrödinger field theory under conformal rescaling demonstrates the rich symmetry structure of the theory. The action is invariant under a set of transformations that form the Schrödinger group, which includes dilations and special conformal transformations. This conformal symmetry has significant implications for the behavior of the system and its applications in various physical contexts. The results of this analysis provide a deeper understanding of the fundamental properties of Schrödinger field theory and its role in modern physics. The insights gained from this study can be extended to more complex systems and interacting theories, contributing to the broader understanding of conformal field theories and their applications in diverse areas of physics, such as condensed matter physics, statistical mechanics, and string theory. The exploration of these symmetries is crucial for advancing our knowledge of the fundamental laws of nature and for developing new theoretical frameworks for understanding the physical world.