Schwinger's variational principle in Einstein$-$Cartan gravity

By applying Schwinger's variational principle to the Einstein-Cartan action, the authors derive quantum commutation relations between the metric and torsion tensors.

The Gist

Imagine the universe as a giant, flexible trampoline. In the classic view of gravity (Einstein's General Relativity), this trampoline is smooth and perfect. If you place a heavy bowling ball on it, the fabric curves down. If you roll a marble nearby, it follows that curve. This is how we usually think of space and time: a smooth, continuous sheet.

But this paper suggests that if we look at this trampoline through the lens of quantum mechanics (the rules that govern tiny particles), the fabric isn't actually smooth at all. It's more like a piece of fabric that can also twist and spin on its own.

Here is a simple breakdown of what the author, Nikodem Poplawski, is doing:

1. The "Rulebook" of Change

In physics, there's a famous rule called the Principle of Least Action. Think of it like a GPS navigation system. If you want to get from Point A to Point B, the universe always chooses the most efficient path. In classical physics, we find this path by saying, "If I wiggle the path just a tiny bit, the total effort shouldn't change."

The author uses a quantum version of this rule, invented by a physicist named Julian Schwinger.

• The Analogy: Imagine you are trying to take a photo of a moving car. In classical physics, you just measure where the car is. In quantum physics, you are measuring the probability of the car being there. Schwinger's rule says that if you wiggle the "path" of the universe, it changes the "photo" (the quantum state) in a very specific, predictable way.

2. The Two Ingredients: The Sheet and the Twist

The author applies this quantum rule to a theory called Einstein-Cartan gravity.

• The Metric (The Sheet): This is the standard gravity we know. It tells us how space stretches and bends (like the trampoline dipping under a weight).
• The Torsion (The Twist): This is the new ingredient. Imagine the trampoline fabric isn't just bending; it's also twisting like a corkscrew. In standard Einstein gravity, this twist is usually ignored or assumed to be zero. But in Einstein-Cartan theory, this twist is a real, independent thing that exists alongside the bending.

3. The Big Discovery: They Are "Dancing Partners"

The author does the math using Schwinger's rule and discovers something surprising about how these two ingredients interact in the quantum world.

He finds that the "Bending" (the metric) and the "Twisting" (torsion) are quantum partners.

• The Analogy: Think of them like a pair of dancers. In classical physics, they can dance separately. But in the quantum world, they are locked in a dance where you can't know exactly how one is moving without affecting the other.
• The Result: The paper derives a "commutation relation." In plain English, this means: You cannot have a universe where the "twist" is zero and the "bending" is perfectly still at the same time.

4. Why This Matters: No Perfect Symmetry

The most exciting part of the conclusion is what this means for the shape of the universe.

• The Old Idea: We often imagine perfect spheres (like stars) or perfect cylinders (like black holes) where gravity is perfectly symmetrical.
• The New Reality: Because the "twist" (torsion) and the "bend" (metric) are quantum partners, the universe cannot be perfectly symmetrical.
  • If you try to make a perfectly round, spinning ball of gravity, the quantum rules force the "twist" to appear.
  • The paper concludes that exact spherical or axial symmetry does not exist in the quantum universe. The universe is inherently "messy" or "jittery" at the smallest scales because of this twist.

5. The "Pop" Instead of a "Bang"

The author mentions that including this "twist" solves some big headaches in cosmology:

• The Big Bang Singularity: In standard physics, the Big Bang is a point where everything is crushed to infinite density (a singularity). It's like a mathematical error.
• The Solution: With this quantum "twist," the universe doesn't crunch to a point. Instead, imagine a spring being squeezed. Eventually, the spring pushes back. The "twist" creates a repulsive force that stops the crunch and makes the universe bounce back out. So, instead of a "Big Bang," it might have been a "Big Bounce."

Summary

This paper takes the rules of quantum mechanics and applies them to a version of gravity that includes "twisting" space. It proves that in the quantum world, space and time are not just a smooth, static sheet. They are a dynamic, twisting fabric where the "bend" and the "twist" are inextricably linked. This link means the universe can never be perfectly symmetrical and suggests that the beginning of the universe was likely a bounce, not a singularity.

Technical Summary

Here is a detailed technical summary of the paper "Schwinger's variational principle in Einstein–Cartan gravity" by Nikodem Pop lawski.

1. Problem Statement

The paper addresses the challenge of deriving quantum commutation relations for the gravitational field within the framework of Einstein–Cartan (EC) gravity. Unlike standard General Relativity (GR), EC gravity treats the affine connection as independent of the metric, allowing for non-zero torsion (the antisymmetric part of the connection). This theory is particularly relevant for describing matter with intrinsic angular momentum (spin).

The specific problem is to determine the fundamental quantum commutation relations between the metric tensor ($g_{\mu\nu}$) and the torsion tensor ($S^\rho_{\mu\nu}$) without resorting to a full, complex quantization of the theory (which faces issues like renormalizability and unitarity). The author seeks to apply Schwinger's variational principle to the EC action to extract these relations directly from the classical action's boundary terms.

2. Methodology

The author employs Schwinger's variational principle, a generalization of Hamilton's principle to quantum mechanics. The core methodology involves the following steps:

• Schwinger's Principle Formulation: The principle states that the variation of the transition amplitude between an initial state $|\alpha_i\rangle$ and a final state $|\alpha_f\rangle$ is related to the variation of the action operator $\delta I$:
  $$\delta \langle \alpha_f | \alpha_i \rangle = \frac{i}{\hbar} \langle \alpha_f | \delta I | \alpha_i \rangle$$
  In the Heisenberg picture, this leads to the operator relation:
  $$\delta O = -\frac{i}{\hbar} [O, \delta I]$$
  where $O$ is an operator and $[A, B]$ is the commutator.

• Application to Field Theory:

  1. The author first reviews the application of this principle to a particle and the electromagnetic field to demonstrate how canonical commutation relations (e.g., $[A_\alpha, E_\beta]$) arise from boundary variations of the action.
  2. The method is then applied to the Einstein–Cartan action. The action $I$ is decomposed into a bulk term (involving the Ricci scalar) and a boundary (hypersurface) term $I_S$.
  3. The EC action is varied with respect to the torsion tensor ($S^\rho_{\mu\nu}$). The variation of the action at the boundary yields a surface integral involving the torsion vector $S_\mu = S^\nu_{\mu\nu}$ and the metric density.

• Derivation of Commutators:
  By substituting the boundary variation $\delta I_S$ into Schwinger's principle (Eq. 2) for the operator $O = S_\mu$, the author derives the condition required for the variation to hold. This leads to specific commutation relations between the torsion vector and the metric components.

3. Key Contributions

• Derivation of Metric-Torsion Commutators: The paper derives the first explicit quantum commutation relations between the metric tensor and the torsion vector in Einstein–Cartan gravity.
• Identification of Conjugate Variables: It establishes that the mixed time-space components of the metric tensor ($g_{\mu 0}$) are canonically conjugate to the torsion vector ($S_\mu$).
• Boundary Term Analysis: The work highlights the physical significance of the hypersurface term in the Einstein–Cartan action, showing that it is not merely a mathematical artifact but carries the quantum information regarding the conjugate momenta of the torsion field.

4. Key Results

The central result is the derivation of the equal-time commutation relations (Eqs. 7 and 8 in the paper):

1. Commutativity of Torsion: The components of the torsion vector commute with each other:
   $$[S_\mu(x, t), S_\nu(x', t)] = 0$$
2. Metric-Torsion Commutation: The torsion vector does not commute with the mixed metric components:
   $$[S_\mu(x, t), g_{\nu 0}(x', t)] = \frac{i}{2\hbar \kappa c} \delta^\nu_\mu \delta(x - x') = 4\pi i l_P^2 \delta^\nu_\mu \delta(x - x')$$
   Where $l_P$ is the Planck length and $\kappa$ is the Einstein gravitational constant.

Physical Implications of Results:

• Non-Vanishing Torsion: Since the commutator is non-zero, the torsion vector $S_\mu$ cannot be zero in a quantum state.
• Non-Vanishing Metric Components: Similarly, the components $g_{\mu 0}$ cannot be zero.
• Absence of Exact Symmetries: Consequently, exact spherically and axially symmetric gravitational fields do not exist in the quantum theory. In classical EC gravity coupled to spinors, the torsion vector often vanishes (leaving only the completely antisymmetric part), but the quantum relation (8) forces the torsion vector to be non-zero even in a vacuum.
• Intrinsic Torsion: Spacetime possesses intrinsic torsion at the quantum level, even in the absence of matter.

5. Significance

• Quantum Structure of Spacetime: The paper suggests that the quantum nature of gravity introduces a fundamental "fuzziness" or intrinsic torsion to spacetime geometry, preventing the existence of perfectly symmetric classical solutions (like the Schwarzschild metric) at the quantum level.
• Ultraviolet Cutoff: The non-vanishing torsion supports the author's previous work suggesting that torsion requires fermions to be spatially extended, potentially providing a natural ultraviolet cutoff in quantum field theory and resolving singularities (like the Big Bang) via a nonsingular bounce.
• Methodological Bridge: The paper demonstrates that Schwinger's variational principle is a powerful tool for extracting quantum commutation relations in gravity without needing a full canonical quantization procedure, which is notoriously difficult for non-renormalizable theories like GR.
• Limitations: The author notes that while these relations are derived at a semi-classical level (treating the action variation as an operator), a full quantization of Einstein–Cartan theory would still need to address standard issues such as renormalizability and unitarity.

In summary, Pop lawski uses Schwinger's principle to show that in Einstein–Cartan gravity, the metric and torsion are quantum mechanically entangled, forcing spacetime to possess intrinsic torsion and forbidding exact classical symmetries in the quantum regime.