Spontaneous Symmetry Breaking Breaks Time-Reversal Symmetry

Jose A. Magpantay

arxiv: 1907.05216 · v1 · pith:GX632NIInew · submitted 2019-07-08 · ⚛️ physics.gen-ph · hep-th

Spontaneous Symmetry Breaking Breaks Time-Reversal Symmetry

Jose A. Magpantay This is my paper

Pith reviewed 2026-05-25 01:01 UTC · model grok-4.3

classification ⚛️ physics.gen-ph hep-th

keywords spontaneous symmetry breakingtime-reversal symmetryarrow of timegauge theoryO(2) symmetryLagrangian formulation

0 comments

The pith

Spontaneous symmetry breaking explicitly breaks time-reversal symmetry in an O(2) gauge theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that spontaneous symmetry breaking, a standard mechanism in particle physics and early-universe evolution, also breaks time-reversal symmetry explicitly. It works in a simplified point-mechanics model with O(2) gauge symmetry by writing the Lagrangian with a time step function that holds for all times. A sympathetic reader would care because this ties the thermodynamic arrow of time directly to the same process that generates masses and other asymmetries after the Big Bang.

Core claim

Spontaneous symmetry breaking explicitly breaks time-reversal symmetry. For simplicity this is shown in a point mechanics gauge theory with symmetry group O(2). The proof relies on expressing the Lagrangian with a time step function that remains valid for any time.

What carries the argument

The time step function inserted into the Lagrangian to make it valid at all times, which exposes the loss of time-reversal invariance once the O(2) symmetry is spontaneously broken.

If this is right

The vacuum after spontaneous symmetry breaking is not invariant under time reversal.
The arrow of time receives a contribution from the same breaking that sets particle masses.
Any theory relying on spontaneous symmetry breaking inherits an explicit time asymmetry at the level of the Lagrangian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Models of the early universe that invoke spontaneous symmetry breaking would need to track an additional source of time asymmetry beyond the usual thermodynamic one.
Experimental searches for time-reversal violation could look for signatures whose scale matches known symmetry-breaking transitions rather than CP violation alone.

Load-bearing premise

The time step function used to write the Lagrangian valid for any time is a legitimate representation of the dynamics that does not itself introduce the time asymmetry being claimed.

What would settle it

An alternative Lagrangian formulation without any time step function that still describes the broken O(2) phase yet preserves time-reversal symmetry under the same transformations.

read the original abstract

The ideas related to the arrow of time are discussed briefly. I then focus on the prevalent physical mechanism in the evolution of the universe and developments in particle physics, spontaneous symmetry breaking, and show that it explicitly breaks time-reversal symmetry. For simplicity, I do this in a point mechanics gauge theory with symmetry group O(2). The proof of breakdown of time-reversal symmetry relies on the use of a time step function to express the Lagrangian valid for any time.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims SSB breaks T symmetry in an O(2) model but the argument rests on a time step function that likely builds in the asymmetry rather than deriving it.

read the letter

The main point is that spontaneous symmetry breaking is said to break time-reversal symmetry, shown via a time step function in the Lagrangian of a simple O(2) point mechanics gauge theory. The abstract presents this as a proof that the mechanism itself introduces time asymmetry. If the step holds up it would link a standard particle physics tool to the arrow of time, but the construction raises immediate questions. The paper does lay out the basic setup clearly in a toy model and keeps the example minimal, which makes the claim easy to follow on its own terms. It also flags the arrow of time discussion upfront before moving to the symmetry breaking step. That structure is direct. The soft spot is exactly the time step function used to write the Lagrangian for any time. This function splits time into distinct regimes around the breaking event, which is the feature that violates T invariance. The abstract gives no indication that the function is forced by the O(2) dynamics or that a fully T-symmetric formulation of the same breaking was considered and ruled out. Without that, the demonstration looks circular: the asymmetry is inserted to make the Lagrangian work across all times and then recovered as a result. Standard treatments of spontaneous symmetry breaking leave the Lagrangian T invariant, so this would need to explain the difference explicitly. The result therefore reduces to showing that an explicitly time-directed Lagrangian breaks T symmetry, which is not a new finding. This paper is aimed at readers working on foundational questions about time and symmetry. A reader wanting a non-circular derivation or a result that engages existing literature on SSB would find little to use. It does not merit sending to peer review because the central technical step does not stand on its own.

Referee Report

2 major / 0 minor

Summary. The paper argues that spontaneous symmetry breaking (SSB) explicitly breaks time-reversal (T) symmetry. It illustrates this in a simplified point-mechanics O(2) gauge theory, claiming that expressing the Lagrangian via a time step function (to ensure validity for any time) demonstrates the T violation induced by SSB.

Significance. If the central claim were established without circularity, it would imply a fundamental link between SSB and the arrow of time, with potential consequences for cosmology and particle physics. However, the construction as described does not appear to deliver a parameter-free or non-circular derivation.

major comments (2)

[Abstract] Abstract: The proof that SSB breaks T symmetry rests on introducing a time step function into the Lagrangian to make it valid for any time. This function partitions time into distinct pre- and post-breaking regimes, which is the precise feature that violates T invariance; the manuscript provides no demonstration that this step function is forced by the O(2) dynamics rather than chosen to encode the desired asymmetry.
[Abstract] Abstract: The claim that SSB 'explicitly breaks' T symmetry contradicts the standard treatment in gauge theories (including O(2) or U(1) models), where the Lagrangian itself remains T-invariant after SSB and any effective T violation arises only from the choice of vacuum or initial conditions, not from the symmetry-breaking mechanism per se.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points that require clarification. We respond to each major comment below and have revised the manuscript to address the concerns raised.

read point-by-point responses

Referee: [Abstract] Abstract: The proof that SSB breaks T symmetry rests on introducing a time step function into the Lagrangian to make it valid for any time. This function partitions time into distinct pre- and post-breaking regimes, which is the precise feature that violates T invariance; the manuscript provides no demonstration that this step function is forced by the O(2) dynamics rather than chosen to encode the desired asymmetry.

Authors: We agree that the original manuscript does not provide an explicit derivation showing that the time step function is required by the O(2) dynamics rather than introduced by hand. The function was used to construct a single Lagrangian expression valid across the symmetry-breaking event at a finite time. We will revise the text to include a more detailed justification of this choice based on the requirement that the Lagrangian describe the full time evolution of the system, including the instant at which the vacuum expectation value is selected. revision: yes
Referee: [Abstract] Abstract: The claim that SSB 'explicitly breaks' T symmetry contradicts the standard treatment in gauge theories (including O(2) or U(1) models), where the Lagrangian itself remains T-invariant after SSB and any effective T violation arises only from the choice of vacuum or initial conditions, not from the symmetry-breaking mechanism per se.

Authors: The manuscript aims to describe the dynamical process of spontaneous symmetry breaking itself, rather than the effective theory after the breaking has occurred. In this dynamical setting the selection of a definite breaking time introduces an explicit distinction between earlier and later times in the Lagrangian. We acknowledge that this differs from the conventional post-breaking treatment in which the Lagrangian remains T-invariant. We will revise the abstract and relevant sections to clarify this distinction and to avoid any implication that the result contradicts standard effective-field-theory results. revision: yes

Circularity Check

1 steps flagged

Time step function in Lagrangian introduces T-asymmetry by construction rather than deriving it from O(2) SSB

specific steps

self definitional [Abstract]
"The proof of breakdown of time-reversal symmetry relies on the use of a time step function to express the Lagrangian valid for any time."

The time step function explicitly divides time into distinct pre- and post-breaking regimes, thereby embedding a preferred time direction. By adopting this representation to write a Lagrangian 'valid for any time,' the T-asymmetry is introduced at the level of the input formalism; the subsequent claim that SSB breaks T symmetry is then true by construction of that formalism rather than derived from the O(2) dynamics.

full rationale

The paper's central result—that spontaneous symmetry breaking in an O(2) gauge theory explicitly breaks time-reversal symmetry—rests on expressing the Lagrangian via a time step function to make it valid for any time. This construction partitions time into before/after regimes, which is the defining feature of T violation. The proof therefore reduces to the representational choice rather than an independent derivation from the symmetry group or dynamics. No alternative T-symmetric Lagrangian formulation is shown to be impossible or ruled out by the SSB itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of expressing the Lagrangian with a time step function; this appears to be an ad-hoc modeling choice introduced to handle any time.

axioms (1)

ad hoc to paper The Lagrangian of the O(2) gauge theory can be expressed using a time step function that remains valid for any time.
The proof of time-reversal symmetry breakdown is stated to rely on this representation.

pith-pipeline@v0.9.0 · 5587 in / 1364 out tokens · 32058 ms · 2026-05-25T01:01:37.978123+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArrowOfTime.lean forward_accumulates / z_monotone_absolute / arrow_from_z unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The complete Lagrangian valid at any time t can be written down as L(t) = Θ(t0 − t)Ls + Θ(t − t0)Lsb, where Θ(t) is the step function... Under time-reversal... the step functions are not [invariant].

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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+.... (26) Substituting equation (24) and carefully tracking the integration t imes from ti to t0 to t when appropriate, equation (26) simpliﬁes into U (ti, t) = exp ( −i ℏ [Hs(t0 − ti) + Hsb(t − t0)] ) , (27) where now the usual 1 n! factors that go with the exponential expansion applies. The solutio n to the Schroedinger equation with the Hamiltonian gi...

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+ (−i ℏ )2 ∫ t ti dt′ 2 ∫ t′ 2 ti dt′ 1H(t′ 2)H(t′

work page

[2] [2]

+ (−i ℏ )3 ∫ t ti dt′ 3 ∫ t′ 3 ti dt′ 2 ∫ t′ 2 ti dt′ 1H(t′ 3)H(t′ 2)H(t′

work page

[3] [3]

+.... (26) Substituting equation (24) and carefully tracking the integration t imes from ti to t0 to t when appropriate, equation (26) simpliﬁes into U (ti, t) = exp ( −i ℏ [Hs(t0 − ti) + Hsb(t − t0)] ) , (27) where now the usual 1 n! factors that go with the exponential expansion applies. The solutio n to the Schroedinger equation with the Hamiltonian gi...

work page

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E., A Modern Course in Statistical Physics, Un iversity of Texas Press, 1980

Reichl, L. E., A Modern Course in Statistical Physics, Un iversity of Texas Press, 1980

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Hanson, A., Regge, T. and Teitelboim, C., Constrained Ha miltonian Systems, Accademia Nazionale dei Lincei, 1976. 8

work page 1976

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A., Microscopic Irreversibility and the H Theorem, International Journal of Modern Physics B, 26, 201 2,

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work page 1989