Quantum Fields in a Big Crunch/Big Bang Spacetime

TL;DR

The paper constructs a unitary quantum field theory across the Big Crunch/Big Bang transition in the compactified Milne spacetime ${\cal M}_C$, showing that free fields undergo essentially unique matching with no particle production, while interactions generate finite tree-level particle production per fixed momentum and can be regulated consistently. It develops multiple complementary formulations, including Minkowski embedding, analytic continuation, and a d-dimensional renormalization framework, to regularize the singularity and fix the vacuum via Hadamard and PT invariance. A formal connection to de Sitter space via a conformal mapping provides an additional lens for understanding mode evolution and vacua (notably the Bunch-Davies vacuum), reinforcing the robustness of the results. Altogether, the work lays groundwork for extending such quantum field theoretic treatments to gravitational backreaction and string/M-theory, potentially yielding a controlled bounce mechanism in early-universe cosmologies.

Abstract

We consider quantum field theory on a spacetime representing the Big Crunch/Big Bang transition postulated in the ekpyrotic or cyclic cosmologies. We show via several independent methods that an essentially unique matching rule holds connecting the incoming state, in which a single extra dimension shrinks to zero, to the outgoing state in which it re-expands at the same rate. For free fields in our construction there is no particle production from the incoming adiabatic vacuum. When interactions are included the total particle production for fixed external momentum is finite at tree level. We discuss a formal correspondence between our construction and quantum field theory on de Sitter spacetime.

Figures (5)

• Figure 1: The compactified Milne universe. On the left is two dimensional Minkowski space. The Lorentz invariant coordinate $t$ satisfying $t^2=T^2-Y^2$ is constant on the dashed surfaces, which provide a spacelike foliation of the causal future and past of the origin. These surfaces are parameterized by a coordinate $y$. Identifying $y$ with $y+L$ compactifies space to produce the spacetime on the right, consisting of two Lorentzian cones joined tip-to-tip at $t=0$. If the circular sections of these cones are orbifolded by a $Z_2$, then the two fixed points of the $Z_2$ are two tensionless branes which collide and pass through one another at $t=0$.
• Figure 2: Our first method for constructing quantum fields on ${\cal M}$, illustrated in a conformal diagram of Minkowski space. The unitary map from the past light cone of the origin, $t=0^-$, to the future light cone $t=0^+$ is defined by free field evolution across the Rindler wedges to the left and right of the origin. Using this rule we obtain a unitary theory on the Milne universe ${\cal M}$, which may then be compactified into the space ${\cal M}_C$ shown in Figure 1.
• Figure 3: Integration contours used to define the positive and negative frequency modes on the entire Milne universe ${\cal M}$.
• Figure 4: Behaviour of the $k_y=0$ modes as they cross $t=0$. With our choice of phase, the imaginary part (Im) diverges logarithmically but the real part (Re) is finite. Analytic continuation along the path shown in Figure 3 causes the real part to be odd in $t$ whereas the imaginary part is even.
• Figure 5: The Milne spacetime is both locally flat (left diagram) and conformally $S^1 \times$ de Sitter (right diagram, with the 5th dimension suppressed). In the de Sitter picture passing through the singularity corresponds to matching future timelike infinity in de Sitter to past timelike infinity.