Event Symmetry in Quantum Models

• Event symmetry in quantum models is the principle where all elementary quantum events—from preparation to measurement—are treated equally, ensuring symmetric transition probabilities.
• Bit symmetry guarantees that any pair of perfectly distinguishable atomic events is reversibly mapped, yielding a self-dual Hilbert space structure typical of quantum mechanics.
• The incorporation of event symmetry constrains quantum models through diverse frameworks, including block-universe, histories-based, and geometric event-based formulations.

Event symmetry in quantum models refers to the principle that all elementary "events"—whether preparation, measurement, or constraint—are to be treated on an equal footing within the mathematical and conceptual frameworks of quantum theory. This requirement emerges at the intersection of generalized probabilistic theories, transition probability frameworks, and modern formulations of quantum histories and quantum logic, imposing powerful structural constraints on possible quantum and post-quantum theories. Event symmetry directly influences the definition and properties of transition probabilities, the foundational object assigned to pairs of quantum events, and distinguishes quantum mechanics from more general non-Hilbertian GPTs. The concept further manifests in block-universe and fixed-point formulations, where it ensures the representational democracy of all spacetime events within quantum evolution.

1. Symmetry Postulates in Quantum Models

Three principal symmetry postulates organize the landscape of probabilistic theories and quantum models featuring distinguished sets of "events" (often identified as atomic projections, effects, or extreme points of convex state spaces):

1. Weak symmetry: Transitivity on pure states; any two atomic events can be interchanged via an automorphism of the state space's order-unit structure.
2. Bit symmetry: Transitivity on orthogonal pairs of atomic events (2-frames); any pair of perfectly distinguishable atoms can be reversibly mapped to any other such pair. This postulate is tightly motivated by the computational role of "qubits" as information carriers.
3. Strong symmetry: Transitivity on k-frames for all $k$ up to the information capacity $m$; any set of $k$ pairwise orthogonal atoms summing to at most the unit can be mapped to any other such set (Niestegge, 2024).

The logical strength grows from weak to strong symmetry. Bit symmetry acts as a critical intermediate case, imposing reversible equivalence between informationally complete 2-level substructures.

2. Transition Probability Framework and Event Symmetry

The transition probability framework refines the GPT paradigm by directly positing the existence of well-defined transition probabilities $\mathbb{P}_e(f)$ between atomic events $e$ and $f$. Within a finite-dimensional compact convex set $\Omega$, whose atomic logic satisfies sharpness and spectrality, the following structures arise:

• Atomic events correspond to the extreme points of the interval $[0, \mathcal{I}]$ in the order-unit space $A_{(\Omega)}$ of affine functionals on $\Omega$.
• Each atom $m$0 supports a unique pure state $m$1 with $m$2 (sharpness).
• The transition probability between atoms $m$3 is defined as $m$4, where $m$5 is the unique state peaking at $m$6.
• These transition probabilities, if invariant under the automorphism group of $m$7, encode event symmetry at the probabilistic level (Niestegge, 2024).

Event symmetry in this context is equivalently the requirement that $m$8 for all atoms $m$9.

3. Bit Symmetry Implies Symmetric Transition Probabilities

A central result establishes that the bit symmetry postulate is both necessary and sufficient for event symmetry in the transition probability framework. Specifically (Niestegge, 2024):

• If $k$0 is bit-symmetric and satisfies sharpness/spectrality, there exists an inner product $k$1 on $k$2 such that

$k$3

with $k$4, ensuring symmetric transition probabilities.

• This construction proceeds by building an automorphism-invariant state and inner product, "twisting" it via the bit symmetry constant $k$5 to achieve self-duality of the positive cone and the desired probabilistic symmetry.

This result shows that event symmetry—the Frobenius symmetry of transition probabilities—is not a primitive axiom but a consequence of bit-reversibility for information pairs. Symmetric transition probabilities are thus tightly linked to reversibility in the abstract state-event structure, and not merely imposed by fiat.

4. Consequences: Classification and Emergence of Jordan Structure

When strong symmetry is imposed in addition to bit symmetry, the allowed quantum models are severely constrained:

• Self-duality and spectral decomposition together guarantee that every state is a convex combination of orthogonal pure states.
• By the Barnum–Hilgert classification, any strongly symmetric, spectrally decomposable, compact convex state space is either:
  • A simplex (classical probability theory), or
  • The state space of a simple Euclidean Jordan algebra (quantum theory over real, complex, or quaternionic Hilbert spaces) (Niestegge, 2024).

Thus, event symmetry---underpinned by bit or strong symmetry---uniquely characterizes the mathematical backbone of standard quantum and classical probabilistic theory.

5. Event Symmetry Beyond Hilbert Spaces: Non-Symmetric Transition Theories

The necessity of event symmetry for quantum mechanics is highlighted by exploring models where transition probabilities are not symmetric:

• In generalized qubit models based on strictly convex, smooth, compact Banach unit balls (other than Hilbert balls), the quantum logical structure remains binary, and sharp, but $k$6 generally (Niestegge, 2022).
• Symmetry holds if and only if the state space is the unit ball of a Hilbert space, i.e., if the underlying order-unit space is a spin factor (a special class of formally real Jordan algebra).
• Models using Banach space dual balls (e.g., $k$7, $k$8) or more exotic geometries (such as the Alfsen–Shultz "triangular pillow") give explicit examples where event symmetry fails due to non-inner-product structure.
• Transition probabilities remain well-defined, but only when symmetry is enforced does the structure collapse to the Hilbertian (spin factor) case (Niestegge, 2022).

This demonstrates that event symmetry is not a generic logical or convex-geometry requirement, but is rather a signature feature of Hilbert-space quantum mechanics.

6. Event Symmetry in Histories-Based and Block-Universe Quantum Models

Recent developments elucidate event symmetry from a temporal and representational standpoint:

• Standard time symmetry (T-symmetry) in quantum mechanics concerns invariance under time reversal in the dynamical laws.
• Event symmetry (in this context) requires that boundary conditions and constraints be imposed at all events on equal footing, eschewing any ontologically privileged initial or final times.
• In fixed-point formulations of quantum theory (using the Keldysh contour), each event corresponds to a constraint gluing the forward and backward time branches, producing symmetric treatment of all events in quantum histories (Ridley et al., 2023).
• The amplitude for any multi-event history is constructed via symmetric insertion of projectors at each event, and statistical rules (such as the Vaidman rule) recover the standard Born rule probabilities.

Under this framework, event symmetry undergirds an "all-at-once," block-universe ontology in which causal and temporal privileging of events is eliminated (Ridley et al., 2023).

7. Event Symmetry and Geometric Event-Based Quantum Mechanics

The geometric event-based (GEB) formulation elevates events to primary status:

• Events serve as primitive elements of the Hilbert space $k$9, with quantum systems emerging as joint amplitudes for these events.
• Full Poincaré symmetry acts as a geometric unitary transformation on the event Hilbert space (Giovannetti et al., 2022).
• Standard quantum mechanics and field theory arise by slicing the event Hilbert space along time, thus recovering conventional dynamical pictures only after a foliation is imposed.

This framework provides an intrinsically symmetric and covariant foundation for quantum models in which event symmetry is hardwired into the kinematics and dynamics at all spacetime points.

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Summary Table: Event Symmetry and Transition Probability Structures

Symmetry Postulate	Transition Probability	Structure Consequence
Weak (pure-state)	No symmetry enforced	Generic spectral convex models
Bit symmetry	Symmetric: $\mathbb{P}_e(f)$0	Self-duality, Hilbertian (spin factor) structure (Niestegge, 2024)
Strong symmetry	Symmetric + full reversibility	Only classical simplices or Euclidean Jordan algebras (quantum)

• Bit symmetry and symmetric transition probabilities: (Niestegge, 2024)
• Event symmetry in block-universe quantum histories: (Ridley et al., 2023)
• Event-based relativistic quantum mechanics: (Giovannetti et al., 2022)
• Non-symmetric transition probabilities in generalized qubit models: (Niestegge, 2022)