Time-Dependent Conformal Transformations

• Time-Dependent Conformal Transformations are maps that preserve the local angle structure while introducing a time-dependent scaling factor, applicable in various physical frameworks.
• They generalize traditional symmetries by allowing metrics and symplectic forms to be rescaled dynamically, thereby extending canonical transformations in both relativistic and non-relativistic contexts.
• These transformations facilitate the reduction of complex, time-dependent systems to solvable forms, with applications ranging from cosmological models and gravitational theories to tensor network simulations in quantum many-body physics.

A time-dependent conformal transformation is a local or global map of spacetime or phase space which preserves the angle structure but introduces a conformal rescaling factor that depends explicitly on a temporal parameter. Such transformations appear in a wide variety of contexts, from quantum field theory and gravitation to Hamiltonian mechanics and tensor network descriptions of many-body systems. The mathematical formalization and physical implications of time-dependent conformal transformations are highly context-dependent: in relativistic settings, they induce Weyl rescalings of the metric with explicit time dependence; in non-relativistic and symplectic structures, they generalize canonical transformations to locally conformal cases with time-dependent conformal factors; in lattice models and quantum information they correspond to position- and time-dependent scaling of quantum states via local tensor operations.

1. Mathematical Definition and General Properties

A conformal transformation is a diffeomorphism $x\mapsto x'$ that rescales the metric locally by a positive function: $g'_{\mu\nu}(x') = \Omega^2(x) g_{\mu\nu}(x)$. In the time-dependent case, the conformal factor $\Omega$ acquires explicit time dependence, i.e., $\Omega = \Omega(x,t)$ or $\Omega = \Omega(\tau)$ (with conformal time $\tau$).

In locally conformal symplectic (LCS) geometry, a time-dependent conformal transformation is a one-parameter family $\Phi_t$ of diffeomorphisms such that for the 2-form $\omega(t)$ and Lee form $\theta$, $\Phi_t^*\omega(t) = \sigma(t)\omega(0)$, with $\sigma(t)$ a positive, time-dependent function and $\Phi_t^*\theta = \theta$ (Azuaje et al., 15 Sep 2025, Ragnisco et al., 2021). The infinitesimal generator $X_t$ of $\Phi_t$ satisfies $L_{X_t}\omega = d(\ln \sigma(t))\wedge\omega$ and $L_{X_t}\theta = 0$.

Time-dependent conformal transformations generalize the notion of symmetry: while strictly canonical transformations preserve the relevant structure (metric, symplectic form) exactly, conformal ones preserve it up to a scale factor. This permits richer dynamical behaviors and accommodates systems with explicit time dependence or varying scale.

2. Time-Dependent Conformal Transformations in Field Theory and Gravity

2.1 Special Conformal Transformations and Relativistic Spacetimes

In relativistic settings, such as general relativity, time-dependent conformal transformations of the metric can be constructed via special conformal transformations (SCT). For example, starting from Minkowski space, the SCT, given by inversion–translation–inversion, leads to a conformally flat metric with time-dependent conformal factor: $$ds^2 = \frac{\eta_{\mu\nu}dx^\mu dx^\nu} {\left[(1-ax)^2 - a^2 t^2\right]^2}$$ where $a$ is a parameter interpreted as an acceleration, and the denominator introduces explicit time dependence (Culetu, 2013). The resulting metric is sourced by an anisotropic fluid with negative energy density and positive pressures, and for small $a$ is related infinitesimally to the Kruskal near-horizon symmetries (Majhi–Padmanabhan generators).

In linearized (static) approximation, the energy-momentum tensor reduces to a cosmological-constant form $T_{\mu\nu} = -\Lambda g_{\mu\nu}$ with $\Lambda \propto a^2$. The full time-dependent metric admits expanding/contracting spherical null horizons, finite curvature invariants for $f(x,t)>0$, and can be generalized by varying $a^\mu$ or by additional dilational maps.

2.2 Time-Dependent Conformal Transformations in Cosmology

Homogeneous cosmological models such as FLRW can be mapped to new coordinates through conformal transformations of the form $\tau \to \tilde{\tau}=f(\tau)$, $$ds^2\to -d\tilde\tau^2 + \tilde{a}^2(\tilde\tau)d\vec{x}^2$$ with the scale factor transforming as $\tilde{a}(\tilde\tau) = h(\tau)^{1/3} a(\tau)$, $h(\tau) = df/d\tau$ (Achour et al., 2020). The action for a free scalar plus gravity,

$$S = \int d\tau \left[ \frac{v \dot{\phi}^2}{2} - \frac{\dot{v}^2}{2v} - \frac{3\Lambda}{4} v \right],$$

is invariant under Möbius transformations of proper time ($SL(2,\mathbb{R})$), with the Schwarzian derivative quantifying the obstruction to full Diff$(S^1)$ invariance.

Performing a time-conformal transformation with constant Schwarzian generates a cosmological constant from the $\Lambda=0$ theory. An extended framework makes $\Lambda(\tau)$ into a conformal gauge field, with transformation law $$\tilde{\Lambda}(f(\tau)) = h^{-2}(\tau) \left[ \Lambda(\tau) + \frac{2}{3}\text{Sch}[f]\right]$$.

3. Time-Dependent Conformal Transformations in Hamiltonian and Symplectic Mechanics

In classical Hamiltonian mechanics, time-dependent conformal (canonoid) transformations generalize canonical transformations to locally conformal symplectic geometries by allowing the two-form $\omega$ to change by a time-dependent factor.

3.1 Structure and Generating Functions

A time-dependent family of diffeomorphisms $\Phi_t$ on $(M,\omega,\theta)$ is conformal if $\Phi_t^*\omega = \sigma(t)\omega$ and $\Phi_t^*\theta = \theta$ (Azuaje et al., 15 Sep 2025, Ragnisco et al., 2021). Locally, the form $\omega$ can be written in "twisted Darboux" coordinates, and generating functions exist (Type I, II, III) with explicit $\sigma(t)$ dependence; for example: $$P_i dQ^i - K dt = \sigma(t) \left[ p_i dq^i - H dt \right] - dF$$ yielding modified Hamilton equations: $$\dot Q^i = \frac{1}{\sigma(t)} \frac{\partial K}{\partial P_i}, \quad \dot P_i = -\frac{1}{\sigma(t)}\frac{\partial K}{\partial Q^i}.$$

The associated Noether-like theorem asserts that any generator $X_t$ with $L_{X_t} \omega = d(\ln \sigma)\wedge\omega$, $L_{X_t}\theta=0$, and $L_{X_t}H=0$ yields a conserved quantity up to the conformal factor.

3.2 Hamilton–Jacobi Formalism and Contact Uplift

The time-dependent Hamilton–Jacobi equation on an LCS manifold acquires an additive term from the Lee form: $$\frac{\partial S}{\partial t} + H(q, \partial_q S, t) + \langle \theta, \partial_q S \rangle = 0$$ (Ragnisco et al., 2021). Locally, LCS dynamics can be uplifted to contact geometry on $M \times \mathbb{R}_t$ via $\alpha = e^{-\sigma(q)}\lambda + dt$, with $d\alpha = e^{-\sigma}\omega$.

4. Applications in Quantum and Statistical Systems

4.1 Tensor Networks as Discrete Conformal Maps

In quantum many-body physics, tensor network architectures such as matrix product operators (MPO), Euclideons, and the multi-scale entanglement renormalization ansatz (MERA) realize non-uniform and time-dependent conformal transformations in lattice models (Milsted et al., 2018).

• Non-uniform Euclidean time evolution is implemented by selecting a strip of sites on the spin chain, embedding "euclideon" tensors $e$ along this strip corresponding to the profile $f(x)$, with smoothers at the strip boundaries. The resulting tensor network $V_{\text{net}}$ approximates the operator $U[f]=\exp[-\int f(x)h(x)dx]$.
• Local scale transformations are built from truncated MERA layers with position-dependent dilation profile $b(x)$, healing dangling bonds with optimized "smoother" tensors.

Benchmarking on the critical Ising chain, the networks reproduce CFT predictions for low-energy state transformations, with errors in tower-off-diagonal elements vanishing as $O(1/N^p)$. The key feature is that a fixed set of tensors suffices to represent arbitrary finite conformal maps on the low-energy sector, independent of lattice size or transformation profile.

4.2 Quadratic Systems and Bargmann Conformal Lifts

Time-dependent conformal transformations provide a framework for mapping time-dependent harmonic oscillators to constant-frequency or free systems via time reparametrization and scaling. The key equations are: $$\xi = \frac{x}{a(t)}, \quad \tau = \int^t \frac{du}{a^2(u)}, \quad \ddot a + \omega^2(t) a = 0$$ mapping the original action to that of a free particle (Zhang et al., 2021, Zhao et al., 2021). The associated Eisenhart–Duval lift leads to two Bargmann spacetimes $ds^2_{\text{osc}}$ and $ds^2_{\text{free}}$ related by a conformal factor $\Omega^2(t) = \dot{\tau}(t)$. The quantum propagator transforms with a Maslov phase determined by the mapping.

Plane gravitational waves in Brinkmann coordinates,

$ds^2 = dx^i dx^i + 2 du dv - H_{ij}(u)x^ix^j du^2$

are likewise related by time-dependent conformal transformations to simpler “Brdička oscillator” metrics, preserving the null geodesics and encoding memory effects.

5. Time-Dependent Conformal Symmetries in Non-Relativistic and Newton–Cartan Geometry

In Newton–Cartan geometry, time-dependent conformal transformations act on the “spatial metric” $\gamma^{\alpha\beta}$ and “absolute clock” $\theta_\alpha$ as $$\phi^*\gamma^{\alpha\beta} = f(x)\gamma^{\alpha\beta}$$, $\phi^*\theta_\alpha = g(x)\theta_\alpha$, with $f$, $g$ arbitrary positive functions (Duval et al., 2016). Time reparametrizations $t \to \tilde{t} = \phi(t)$ and $x^i \to \tilde{x}^i = \Omega(t) x^i$ are allowed, as are independent space- and time-dilations.

The conformal–Newton–Cartan group is 13-dimensional (rotations, translations, boosts, two dilations), and its Bargmann extension gains another (non-central) generator. Specific solutions (Newtonian cosmology, Newton–Hooke) admit explicit time-dependent conformal maps via the Eisenhart lift, e.g.,

$$(x, t, s) \mapsto \left( \frac{x}{\Theta(t)}, \int \frac{dt}{\Theta^2(t)}, s + \frac{1}{2} \frac{\dot{\Theta}}{\Theta} |x|^2 \right)$$

with $\Theta(t)$ solving the cosmological field equations. The Schwarzian derivative of the time reparametrization is directly related to the matter density, showing deep links between conformal geometry and Newtonian cosmological evolution.

6. Physical Implications and Theoretical Consequences

Time-dependent conformal transformations underpin analytical techniques across quantum, classical, and cosmological systems. They:

• Allow the transfer of solutions and symmetries between different dynamical systems (e.g., mapping oscillator problems to free particle dynamics by time-dependent scaling).
• Enable efficient tensor network representations of critical quantum states, facilitating studies of entanglement and RG flows (Milsted et al., 2018).
• Provide mechanisms to generate effective cosmological constants, anisotropic fluids, or accelerated expansion in gravitational theories by suitable choice of conformal factor (Culetu, 2013, Achour et al., 2020, Çiftci et al., 2019).
• Generalize the class of admissible dynamics and conserved quantities in Hamiltonian and symplectic mechanics, extending classical Noether theorems (Azuaje et al., 15 Sep 2025).
• Clarify the role of time in cosmological observables, leading to de-parameterized relational approaches and highlighting gauge redundancies in standard treatments (Achour et al., 2020).

A plausible implication is that many analytically intractable time-dependent systems can be reduced to solvable forms via conformal transformations, provided the conformal structure of the underlying theory is appropriately exploited.

7. Representative Table: Types and Domains of Time-Dependent Conformal Transformations

Domain	Transformation Form	Key Effects
Relativistic Gravity	$g'_{\mu\nu}(x) = \Omega^2(x,t) g_{\mu\nu}(x)$	Generates new spacetimes, possibly non-trivial sources
Quantum Many-Body	Tensor network layers with local time/space profile	Implements local RG/time evolution, simulates CFT transformation
Hamiltonian Mechanics (LCS)	$\Phi_t^*\omega = \sigma(t)\omega$	Enlarges symmetry group, modifies Hamiltonian structure
Non-Relativistic (Newton–Cartan)	$t \to \tilde{t} = \phi(t),\ x^i \to \Omega(t)x^i$	Realizes independent scaling of time and space

These applications demonstrate that the theoretical apparatus of time-dependent conformal transformations is a unifying principle connecting disparate areas of physics—geometric structures, dynamical systems, and quantum lattice models—through their capacity to relate, classify, and simplify systems by scale-respecting maps with explicit time dependence.