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A Hilbert-Space Analogy for Sexual Orientation and Gender Identity

This paper introduces a finite-dimensional Hilbert-space analogy using Dirac notation to mathematically model bisexual attraction and gender identity trajectories, while explicitly clarifying that the framework is a conceptual tool rather than a physical theory or measurement procedure for human sexuality.

Original authors: Sofia Fatigoni

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Sofia Fatigoni

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Human beings are complex, and the ways we understand who we are and who we love often resist simple labels. In the social sciences, researchers frequently look for models that can capture this complexity without reducing a person to a single box. One such approach borrows a framework from physics, specifically the mathematics used to describe the behavior of subatomic particles. This field, known as quantum mechanics, deals with states that can exist in multiple possibilities at once until they are observed. While the behavior of electrons is governed by physical laws, the researchers behind a new study have proposed using the same mathematical language as a formal analogy to better understand human sexual orientation and gender identity. The goal is not to claim that people are made of quantum particles, but to use the logic of these equations to clarify a common social misunderstanding: that seeing someone in a relationship tells us everything about who they are.

The core of this work, led by Sofia Fatigoni at the California Institute of Technology, is a proposal to represent a person's attractions using a system of weighted possibilities rather than fixed categories. In the simplest version of this model, a person's sexual orientation is described as a combination of two distinct possibilities: attraction to the same gender and attraction to a different gender. The researchers assign a specific weight to each possibility, much like balancing a scale. If a person is exclusively attracted to one gender, the weight for that side is at its maximum, and the other is zero. However, for a bisexual person, both weights are present simultaneously. The model shows that these weights do not have to be equal; a person can feel a stronger pull toward one gender than the other and still be bisexual. The mathematics of this system ensures that the total of these weights always adds up to a whole, representing the complete picture of that person's capacity for attraction.

A crucial finding of the study is what happens when an observer sees a single relationship. If a bisexual person is seen dating someone of the opposite gender, an observer might mistakenly conclude that the person is now exclusively attracted to that gender, or that the attraction to their own gender has vanished. The model demonstrates that this is a logical error. Observing one relationship is like seeing only one outcome of a measurement; it does not erase the other possibilities that remain part of the person's full identity. The person's orientation remains a combination of both possibilities, even if their current partner represents only one of them. The researchers emphasize that a relationship does not "collapse" a person's identity into a single state, nor does it erase other attractions. To label someone as straight or gay solely based on their current partner is to mistake a single observed event for the entire person.

The model also expands to include more than just two gender categories. The researchers introduce a third possibility to account for attraction to people who exist outside the traditional binary of man and woman, such as nonbinary individuals. In this extended view, a person's attraction is a blend of three possibilities: same gender, different gender, and attraction to genders outside that binary. The study clarifies that this does not mean attraction to nonbinary people is exclusive to one specific label like pansexuality, nor does it suggest that nonbinary people are a single, uniform third category. Instead, the model treats these as coordinates in a space of possibilities, allowing for a more nuanced description of how attraction can span across different gender identities.

Just as the paper separates the question of who a person is attracted to from the question of who they are, it treats gender identity as a separate space entirely. A person's gender identity is their own internal sense of self, which is distinct from the genders they are attracted to. The researchers use the same mathematical style to describe a nonbinary identity as a blend of identifying as a man and identifying as a woman, though they note that this is just one possible way to describe it. Some people may identify with neither, or use entirely different terms. The model also allows for time to be included, showing how a person's sense of self might shift over time. For example, a person might start with a strong sense of one identity and gradually move toward another, with the mathematical weights changing smoothly over a period of time. This is presented as a way to visualize a personal journey, not as a universal rule for how all transgender people experience their lives.

The author is careful to state that this is an analogy, not a physical theory about human biology. The mathematics does not predict how often people will form relationships, because real-life choices depend on many factors like opportunity, preference, and circumstance. No one is performing a controlled experiment on a person's identity, and there is no physical collapse of a wave function happening in human interaction. The value of the work lies in the logical constraint it provides: it proves that you cannot know a person's full orientation or identity from a single observation. The validity of who a person is rests on their own self-description, not on an external label assigned by an observer based on a single snapshot of their life. By using this formal structure, the study offers a clear way to understand that human identity is a full, complex state that cannot be reduced to a single data point.

Technical Summary: A Hilbert-Space Analogy for Sexual Orientation and Gender Identity

Problem Statement
The paper addresses the conceptual challenge of modeling sexual orientation and gender identity in a way that captures their complexity without reducing individuals to single observed behaviors. Specifically, it seeks to formalize the distinction between a person's internal state (orientation and identity) and the limited information available from a single observed relationship or interaction. The core problem is the logical fallacy of inferring a person's entire orientation vector from a single "measurement" (e.g., observing a bisexual person with a partner of a specific gender), which often leads to misclassification (e.g., labeling a bisexual person as straight or gay based solely on their current partner).

Methodology
The author employs a formal mathematical analogy using finite-dimensional Hilbert spaces and Dirac notation, typically associated with quantum mechanics, to construct a state-space representation. The methodology is strictly defined as an analogy, not a physical theory or an empirical measurement procedure for human sexuality.

Orientation State Space:
- A two-state basis is established using orthonormal vectors $|S\rangle$ (attraction to the same gender) and $|D\rangle$ (attraction to a different gender).
- A "bisexual" state is represented as a superposition: $|\psi_{bi}\rangle = \alpha |S\rangle + \beta |D\rangle$ , where $\alpha$ and $\beta$ are probability amplitudes.
- The state is normalized such that $|\alpha|^2 + |\beta|^2 = 1$ , where the squared magnitudes represent the probability weights of projection outcomes, not the frequency of relationships.
- The model is extended to a three-state basis ( $|S\rangle, |D\rangle, |O\rangle$ ) to include attraction to genders outside the binary ( $|O\rangle$ ), represented as $|\psi_3\rangle = \alpha |S\rangle + \beta |D\rangle + \gamma |O\rangle$ .
Identity State Space:
- A distinct Hilbert space is defined for gender identity, separate from the orientation space.
- A nonbinary identity is modeled as a superposition of basis states $|M\rangle$ (man) and $|W\rangle$ (woman): $|\chi_{NB}\rangle = u |M\rangle + v |W\rangle$ .
- A time-dependent trajectory is introduced to model a specific account of gender transition or evolving self-understanding: $|\chi(t)\rangle = e^{-t/\tau} |M\rangle + \sqrt{1 - e^{-2t/\tau}} |W\rangle$ . This ensures the state remains normalized at all times $t$ .
Tensor Product Structure:
- The paper posits that identity and orientation exist in independent spaces ( $H_{identity}$ and $H_{orientation}$ ). The combined state is represented as a tensor product $|\chi\rangle \otimes |\psi\rangle$ , emphasizing that specifying one factor does not logically determine the other.

Key Contributions and Results

Formalization of Bisexuality: The paper provides a mathematical framework where bisexuality is not a 50/50 split but a normalized superposition of states with arbitrary weights ( $\alpha, \beta$ ). It demonstrates that exclusive orientations (gay/straight) are merely limiting cases where one coefficient is zero.
Inference Constraints: The analysis of "one observed eigenstate" demonstrates that observing a relationship (a projection onto $|S\rangle$ or $|D\rangle$ ) classifies the relationship, not the person. A single observation does not collapse the person's full orientation vector; the full state $|\psi_{bi}\rangle$ remains valid and uncollapsed.
Distinction of Spaces: By separating orientation and identity into distinct vector spaces, the model formally illustrates that gender identity does not dictate sexual orientation, and vice versa.
Handling Non-Binary and Trans Identities: The model clarifies that nonbinary identity is not a "third gender" eigenstate but can be a superposition of existing basis states. Similarly, it explicitly states that "trans" is not a third gender eigenstate; a trans man is represented within the $|M\rangle$ basis, not as a separate category.

Significance and Claims
The paper explicitly claims its significance as a formal analogy rather than a physical model.

Limitations: The author states that the amplitudes are formal coordinates, not empirical frequencies. The mathematics entails no physical collapse, interference, or quantum dynamics. Real-life partner choice involves opportunity, preference, and circumstance, which are not captured by the projection operators.
Core Argument: The primary utility of the model is to establish an inference constraint: one observed relationship cannot specify a person's full orientation.
Conclusion: The validity of a person's identity and orientation rests on their self-description, not on an externally assigned state vector derived from limited observations. The model serves to illustrate the information omitted by single observations and to provide a structured way to discuss attraction across gender categories without overlap.

Technical Summary: A Hilbert-Space Analogy for Sexual Orientation and Gender Identity

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