[Paper Review] Future Dependent Initial Conditions from Imaginary Part in Lagrangian

H. B. Nielsen, Masao Ninomiya|ArXiv.org|Dec 5, 2006

International Science and Diplomacy6 references14 citations

TL;DR

This paper proposes a complex action formalism in quantum field theory by introducing an imaginary part in the Lagrangian, which dynamically determines initial conditions—particularly in the early universe—via path integral weighting. The model predicts a significantly broader Higgs boson width than standard model calculations due to non-unitary effects, suggesting potential experimental anomalies at the LHC.

ABSTRACT

We want to unify usual equation of motion laws of nature with "laws" about initial conditions, second law of thermodynamics, cosmology. By introducing an imaginary part -- of a similar form but different parameters as the usual real part -- for the action to be used in the Feynmann path way integral we obtain a model determining (not only equations of motion but) also the initial conditions, for say a quantum field theory. We set up the formalism for e.g. expectation values, classical approximation in such a model and show that provided the imaginary part gets unimportant except in the Big Bang era the model can match the usual theory. Speculatively requiring that there be place for Dirac strings and thus in principle monopoles in the model we can push away the effects of the imaginary part to be involved only with particles not yet found. Most promising for seeing the initial condition determining effects from the imaginary part is thus the Higgs particle. We predict that the width of the Higgs particle shall likely turn out to be (appreciably perhaps) broader than calculated by summing usual decay rates. Higgs machines will be hit by bad luck.

Motivation & Objective

• To unify equations of motion with initial condition selection within a single theoretical framework.
• To address the cosmological and thermodynamic problem of initial conditions by embedding them in the action via an imaginary component.
• To explore whether non-unitary dynamics from a complex action can suppress prearranged future configurations, avoiding time-asymmetry paradoxes.
• To provide a quantum field theory framework where initial conditions are not postulated but derived from the action's imaginary part.
• To predict observable deviations in Higgs boson decay width due to non-unitary contributions from the imaginary Lagrangian.

Proposed method

• Extend the standard real action $ S = S_R $ to a complex action $ S = S_R + iS_I $, where $ S_I $ is derived from a separate Lagrangian density $ \mathcal{L}_I $ with distinct parameters.
• Use the Feynman path integral with complex action to compute transition amplitudes and expectation values, replacing the standard unitary $ S $-matrix.
• Formulate a modified transition probability using a double integral over boundary conditions: $ \sum_{i,f} \left| \int_{\text{boundary}: i,f} e^{iS[\phi]} \mathscr{D}\phi \right|^2 $, normalized by the unprojected amplitude.
• Introduce projection operators $ P_{\mathcal{O}_i \in \bar{M}_i} $ into the path integral to define probabilities for specific observables at different times.
• Derive a new transition amplitude formula: $ \text{Prod}(|\psi_f\rangle, |\psi\rangle) = \frac{|\langle\psi_f|S|\psi\rangle|^2 \langle\psi_f|\rho_{f\text{ from }t_f}|\psi_f\rangle}{\langle\psi|S^\dagger S|\psi\rangle} $, which incorporates non-unitary evolution.
• Apply the formalism to the Higgs sector, showing that the imaginary part suppresses Higgs production in the early universe, leading to a predicted broadening of the Higgs width.

Experimental results

Research questions

• RQ1 Can initial conditions in quantum field theory be dynamically determined by an imaginary part in the action, rather than being postulated?
• RQ2 How does the inclusion of a complex action affect the unitarity and probability interpretation of the path integral?
• RQ3 Why are prearranged future configurations (e.g., 'God's hand' events) not observed in nature, despite time-symmetric dynamics?
• RQ4 What constraints on the imaginary Lagrangian are needed to suppress prearrangements while still fixing initial conditions?
• RQ5 Can the imaginary part of the action lead to measurable deviations in Higgs boson decay properties, such as width?

Key findings

• The imaginary part of the action, $ S_I $, acts as a selection mechanism in the path integral, favoring specific paths and thus determining initial conditions dynamically.
• The model predicts that the Higgs boson width will be appreciably broader than standard model predictions due to non-unitary contributions from the imaginary Lagrangian.
• Classical approximations of the model show that homogeneous terms like the Dirac Lagrangian vanish modulo total divergences when equations of motion are imposed, even if $ \mathcal{L}_I \neq 0 $, preserving consistency.
• The formalism suppresses prearranged configurations by requiring consistency of Dirac strings, which may explain the absence of observable future-organized events.
• The model suggests that LHC experiments may face unexpected failures or reduced luminosity due to a suppression of Higgs production in the early universe, implying 'bad luck' at the accelerator.
• The derived transition amplitude formula $ \text{Prod}(|\psi_f\rangle, |\psi\rangle) $ generalizes the standard $ S $-matrix and incorporates non-unitary evolution through a density matrix at the final time.