God does not play dice with the universe.
—Albert Einstein
Einstein, stop telling God what to do.
—Niels Bohr (attributed)
To understand precisely why quantum mechanics offers no refuge for free will, we must examine its mathematical structure with care. We present here the axiomatic foundations of the theory, with definitions, postulates, and theorems stated precisely and proved where appropriate.11
Definition 2.1 (Inner Product Space). An inner product space over \(\mathbb{C}\) is a vector space \(V\) over \(\mathbb{C}\) equipped with a map \(\langle \cdot | \cdot \rangle : V \times V \to \mathbb{C}\) (called the inner product) satisfying for all \(|\phi\rangle, |\psi\rangle, |\chi\rangle \in V\) and all \(\alpha, \beta \in \mathbb{C}\):
Conjugate symmetry: \(\langle \phi | \psi \rangle = \overline{\langle \psi | \phi \rangle}\)
Linearity in the second argument: \(\langle \phi | \alpha\psi + \beta\chi \rangle = \alpha\langle \phi | \psi \rangle + \beta\langle \phi | \chi \rangle\)
Positive definiteness: \(\langle \psi | \psi \rangle \geq 0\), with equality iff \(|\psi\rangle = 0\)
Remark 2.2. We use Dirac’s “bra-ket” notation: \(|\psi\rangle\) denotes a vector (“ket”) in the space, while \(\langle\phi|\) denotes the corresponding linear functional (“bra”). The inner product is written \(\langle\phi|\psi\rangle\).12
Definition 2.3 (Norm and Distance). The norm induced by an inner product is \(\||\psi\rangle\| := \sqrt{\langle \psi | \psi \rangle}\). The distance between \(|\phi\rangle\) and \(|\psi\rangle\) is \(d(|\phi\rangle, |\psi\rangle) := \||\phi\rangle - |\psi\rangle\|\).
Theorem 2.4 (Cauchy-Schwarz Inequality). For any \(|\phi\rangle, |\psi\rangle\) in an inner product space: \[|\langle \phi | \psi \rangle|^2 \leq \langle \phi | \phi \rangle \cdot \langle \psi | \psi \rangle \tag{2.1}\] with equality if and only if \(|\phi\rangle\) and \(|\psi\rangle\) are linearly dependent.
Proof. If \(|\psi\rangle = 0\), both sides are zero and the inequality holds trivially. Assume \(|\psi\rangle \neq 0\).
For any \(\lambda \in \mathbb{C}\), consider the vector \(|\phi\rangle - \lambda|\psi\rangle\). By positive definiteness: \[0 \leq \langle \phi - \lambda\psi | \phi - \lambda\psi \rangle = \langle \phi | \phi \rangle - \lambda\langle \psi | \phi \rangle - \overline{\lambda}\langle \phi | \psi \rangle + |\lambda|^2\langle \psi | \psi \rangle \tag{2.2}\]
Choose \(\lambda = \frac{\langle \phi | \psi \rangle}{\langle \psi | \psi \rangle}\) (well-defined since \(|\psi\rangle \neq 0\)). Substituting: \[\begin{align*} 0 &\leq \langle \phi | \phi \rangle - \frac{\langle \phi | \psi \rangle}{\langle \psi | \psi \rangle}\langle \psi | \phi \rangle - \frac{\overline{\langle \phi | \psi \rangle}}{\langle \psi | \psi \rangle}\langle \phi | \psi \rangle + \frac{|\langle \phi | \psi \rangle|^2}{\langle \psi | \psi \rangle^2}\langle \psi | \psi \rangle \\ &= \langle \phi | \phi \rangle - \frac{|\langle \phi | \psi \rangle|^2}{\langle \psi | \psi \rangle} - \frac{|\langle \phi | \psi \rangle|^2}{\langle \psi | \psi \rangle} + \frac{|\langle \phi | \psi \rangle|^2}{\langle \psi | \psi \rangle} \\ &= \langle \phi | \phi \rangle - \frac{|\langle \phi | \psi \rangle|^2}{\langle \psi | \psi \rangle} \end{align*} \tag{2.3}\]
Rearranging: \(|\langle \phi | \psi \rangle|^2 \leq \langle \phi | \phi \rangle \cdot \langle \psi | \psi \rangle\).
Equality holds iff \(|\phi\rangle - \lambda|\psi\rangle = 0\), i.e., iff \(|\phi\rangle\) and \(|\psi\rangle\) are linearly dependent. ◻
Definition 2.5 (Hilbert Space). A Hilbert space \(\mathcal{H}\) is an inner product space that is complete: every Cauchy sequence in \(\mathcal{H}\) converges to an element of \(\mathcal{H}\).13
Remark 2.6. For finite-dimensional systems (such as spin-\(\frac{1}{2}\) particles), \(\mathcal{H} = \mathbb{C}^n\) suffices. For a single particle moving in three-dimensional space, \(\mathcal{H} = L^2(\mathbb{R}^3)\), the space of square-integrable complex-valued functions on \(\mathbb{R}^3\) with inner product: \[\langle \phi | \psi \rangle = \int_{\mathbb{R}^3} \overline{\phi(\mathbf{x})}\psi(\mathbf{x}) \, d^3x \tag{2.4}\] For the entire universe, \(\mathcal{H}\) would be an enormously complicated tensor product of such spaces for all particles and fields.14
Definition 2.7 (Linear Operator). A linear operator \(\hat{A}\) on \(\mathcal{H}\) is a map \(\hat{A}: \mathcal{D}(\hat{A}) \to \mathcal{H}\), where \(\mathcal{D}(\hat{A}) \subseteq \mathcal{H}\) is the domain of \(\hat{A}\), satisfying: \[\hat{A}(\alpha|\psi_1\rangle + \beta|\psi_2\rangle) = \alpha\hat{A}|\psi_1\rangle + \beta\hat{A}|\psi_2\rangle \tag{2.5}\] for all \(|\psi_1\rangle, |\psi_2\rangle \in \mathcal{D}(\hat{A})\) and \(\alpha, \beta \in \mathbb{C}\).
Definition 2.8 (Adjoint Operator). The adjoint \(\hat{A}^\dagger\) of a linear operator \(\hat{A}\) is defined by: \[\langle \phi | \hat{A}\psi \rangle = \langle \hat{A}^\dagger\phi | \psi \rangle \tag{2.6}\] for all \(|\phi\rangle \in \mathcal{D}(\hat{A}^\dagger)\) and \(|\psi\rangle \in \mathcal{D}(\hat{A})\).15
Definition 2.9 (Hermitian and Self-Adjoint Operators). An operator \(\hat{A}\) is Hermitian if \(\langle \phi | \hat{A}\psi \rangle = \langle \hat{A}\phi | \psi \rangle\) for all \(|\phi\rangle, |\psi\rangle \in \mathcal{D}(\hat{A})\).
An operator \(\hat{A}\) is self-adjoint if \(\hat{A} = \hat{A}^\dagger\) and \(\mathcal{D}(\hat{A}) = \mathcal{D}(\hat{A}^\dagger)\).16
Definition 2.10 (Eigenvalue and Eigenvector). A scalar \(\lambda \in \mathbb{C}\) is an eigenvalue of \(\hat{A}\) if there exists a nonzero vector \(|\lambda\rangle \in \mathcal{D}(\hat{A})\) such that: \[\hat{A}|\lambda\rangle = \lambda|\lambda\rangle \tag{2.7}\] The vector \(|\lambda\rangle\) is called an eigenvector (or eigenstate) corresponding to \(\lambda\).
Theorem 2.11 (Properties of Hermitian Operators). Let \(\hat{A}\) be a Hermitian operator on a Hilbert space \(\mathcal{H}\). Then:
All eigenvalues of \(\hat{A}\) are real.
Eigenvectors corresponding to distinct eigenvalues are orthogonal.
Proof. (1) Reality of eigenvalues: Let \(\hat{A}|\lambda\rangle = \lambda|\lambda\rangle\) with \(|\lambda\rangle \neq 0\). Then: \[\begin{align*} \lambda\langle \lambda|\lambda\rangle &= \langle \lambda|\hat{A}|\lambda\rangle \\ &= \langle \hat{A}\lambda|\lambda\rangle \quad \text{(since $\hat{A}$ is Hermitian)} \\ &= \overline{\langle \lambda|\hat{A}\lambda\rangle} \quad \text{(by conjugate symmetry)} \\ &= \overline{\lambda\langle \lambda|\lambda\rangle} \\ &= \overline{\lambda}\langle \lambda|\lambda\rangle \end{align*} \tag{2.8}\] Since \(|\lambda\rangle \neq 0\), we have \(\langle \lambda|\lambda\rangle > 0\), so \(\lambda = \overline{\lambda}\), meaning \(\lambda \in \mathbb{R}\).
(2) Orthogonality: Let \(\hat{A}|\lambda\rangle = \lambda|\lambda\rangle\) and \(\hat{A}|\mu\rangle = \mu|\mu\rangle\) with \(\lambda \neq \mu\). Then: \[\begin{align*} \lambda\langle \mu|\lambda\rangle &= \langle \mu|\hat{A}|\lambda\rangle \\ &= \langle \hat{A}\mu|\lambda\rangle \quad \text{(since $\hat{A}$ is Hermitian)} \\ &= \overline{\mu}\langle \mu|\lambda\rangle \\ &= \mu\langle \mu|\lambda\rangle \quad \text{(since $\mu$ is real by part 1)} \end{align*} \tag{2.9}\] Thus \((\lambda - \mu)\langle \mu|\lambda\rangle = 0\). Since \(\lambda \neq \mu\), we have \(\langle \mu|\lambda\rangle = 0\). ◻
Theorem 2.12 (Spectral Theorem (Finite-Dimensional Case)). Let \(\hat{A}\) be a Hermitian operator on a finite-dimensional Hilbert space \(\mathcal{H}\) with \(\dim(\mathcal{H}) = n\). Then there exists an orthonormal basis \(\{|a_1\rangle, |a_2\rangle, \ldots, |a_n\rangle\}\) of \(\mathcal{H}\) consisting of eigenvectors of \(\hat{A}\): \[\hat{A}|a_i\rangle = a_i|a_i\rangle, \quad \langle a_i|a_j\rangle = \delta_{ij} \tag{2.10}\] where the eigenvalues \(a_i\) are real. Moreover, \(\hat{A}\) admits the spectral decomposition: \[\hat{A} = \sum_{i=1}^{n} a_i |a_i\rangle\langle a_i| \tag{2.11}\]
Proof. We prove by induction on \(n = \dim(\mathcal{H})\).
Base case (\(n = 1\)): Any nonzero vector is an eigenvector of any operator, and it can be normalized.
Inductive step: Assume the theorem holds for spaces of dimension \(n-1\). Let \(\mathcal{H}\) have dimension \(n\).
Every operator on a finite-dimensional complex vector space has at least one eigenvalue (since the characteristic polynomial has at least one root in \(\mathbb{C}\)). Let \(a_1\) be an eigenvalue of \(\hat{A}\) with normalized eigenvector \(|a_1\rangle\).
Let \(\mathcal{H}' = \{|\psi\rangle \in \mathcal{H} : \langle a_1|\psi\rangle = 0\}\) be the orthogonal complement of \(\text{span}\{|a_1\rangle\}\). Then \(\dim(\mathcal{H}') = n - 1\).
Claim: \(\hat{A}\) maps \(\mathcal{H}'\) to itself. Proof of claim: Let \(|\psi\rangle \in \mathcal{H}'\). Then: \[\langle a_1|\hat{A}\psi\rangle = \langle \hat{A}a_1|\psi\rangle = a_1\langle a_1|\psi\rangle = 0 \tag{2.12}\] so \(\hat{A}|\psi\rangle \in \mathcal{H}'\).
The restriction \(\hat{A}|_{\mathcal{H}'}\) is Hermitian on the \((n-1)\)-dimensional space \(\mathcal{H}'\). By the inductive hypothesis, there is an orthonormal basis \(\{|a_2\rangle, \ldots, |a_n\rangle\}\) of \(\mathcal{H}'\) consisting of eigenvectors of \(\hat{A}\).
Then \(\{|a_1\rangle, |a_2\rangle, \ldots, |a_n\rangle\}\) is an orthonormal basis of \(\mathcal{H}\) consisting of eigenvectors of \(\hat{A}\).
For the spectral decomposition, note that for any \(|\psi\rangle \in \mathcal{H}\): \[|\psi\rangle = \sum_{i=1}^{n} |a_i\rangle\langle a_i|\psi\rangle = \sum_{i=1}^{n} \langle a_i|\psi\rangle |a_i\rangle \tag{2.13}\] Therefore: \[\hat{A}|\psi\rangle = \sum_{i=1}^{n} \langle a_i|\psi\rangle \hat{A}|a_i\rangle = \sum_{i=1}^{n} \langle a_i|\psi\rangle a_i|a_i\rangle = \sum_{i=1}^{n} a_i |a_i\rangle\langle a_i|\psi\rangle = \left(\sum_{i=1}^{n} a_i |a_i\rangle\langle a_i|\right)|\psi\rangle \tag{2.14}\] ◻
Definition 2.13 (Unitary Operator). An operator \(\hat{U}\) is unitary if \(\hat{U}^\dagger\hat{U} = \hat{U}\hat{U}^\dagger = \hat{I}\), where \(\hat{I}\) is the identity operator.17
Theorem 2.14 (Properties of Unitary Operators). Let \(\hat{U}\) be a unitary operator. Then:
\(\hat{U}\) preserves norms: \(\|\hat{U}|\psi\rangle\| = \||\psi\rangle\|\)
\(\hat{U}\) preserves inner products: \(\langle \hat{U}\phi|\hat{U}\psi\rangle = \langle \phi|\psi\rangle\)
All eigenvalues of \(\hat{U}\) have absolute value 1
Proof. (1) and (2): \(\langle \hat{U}\phi|\hat{U}\psi\rangle = \langle \phi|\hat{U}^\dagger\hat{U}|\psi\rangle = \langle \phi|\hat{I}|\psi\rangle = \langle \phi|\psi\rangle\). Setting \(|\phi\rangle = |\psi\rangle\) gives (1).
(3): Let \(\hat{U}|\lambda\rangle = \lambda|\lambda\rangle\) with \(|\lambda\rangle \neq 0\). Then: \[\langle \lambda|\lambda\rangle = \langle \hat{U}\lambda|\hat{U}\lambda\rangle = \langle \lambda|\hat{U}^\dagger\hat{U}|\lambda\rangle = |\lambda|^2\langle \lambda|\lambda\rangle \tag{2.15}\] Since \(\langle \lambda|\lambda\rangle > 0\), we have \(|\lambda|^2 = 1\), so \(|\lambda| = 1\). ◻
We now state the fundamental postulates of quantum mechanics.18
Postulate 2.15 (State Space). The state of a quantum system is represented by a unit vector \(|\psi\rangle\) in a complex Hilbert space \(\mathcal{H}\), called the state space of the system. Two vectors that differ only by a global phase factor \(e^{i\theta}\) represent the same physical state.19
Postulate 2.16 (Observables). Every observable physical quantity \(A\) is represented by a self-adjoint operator \(\hat{A}\) on \(\mathcal{H}\). The possible outcomes of measuring \(A\) are the eigenvalues of \(\hat{A}\).20
Postulate 2.17 (Born Rule). If a system is in state \(|\psi\rangle\) and observable \(\hat{A}\) is measured, the probability of obtaining eigenvalue \(a_i\) is: \[P(a_i) = |\langle a_i|\psi\rangle|^2 \tag{2.16}\] where \(|a_i\rangle\) is the normalized eigenstate corresponding to \(a_i\). If \(a_i\) is degenerate (has multiple linearly independent eigenstates), then: \[P(a_i) = \sum_{j} |\langle a_i^{(j)}|\psi\rangle|^2 = \langle \psi|\hat{P}_i|\psi\rangle \tag{2.17}\] where \(\hat{P}_i\) is the projection operator onto the eigenspace of \(a_i\).21
Postulate 2.18 (Collapse). Immediately after a measurement of observable \(\hat{A}\) yields result \(a_i\), the system is in an eigenstate of \(\hat{A}\) with eigenvalue \(a_i\): \[|\psi\rangle \xrightarrow{\text{measurement}} |a_i\rangle \tag{2.18}\] (or a normalized vector in the eigenspace of \(a_i\) if degenerate).22
Postulate 2.19 (Time Evolution). Between measurements, the state evolves according to the Schrödinger equation: \[i\hbar \frac{d}{dt}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle \tag{2.19}\] where \(\hat{H}\) is the Hamiltonian (total energy operator) of the system and \(\hbar\) is the reduced Planck constant.23
Theorem 2.20 (Unitary Time Evolution). The time evolution given by the Schrödinger equation is unitary. Specifically, if \(\hat{H}\) is time-independent, then: \[|\psi(t)\rangle = \hat{U}(t)|\psi(0)\rangle, \quad \text{where } \hat{U}(t) = e^{-i\hat{H}t/\hbar} \tag{2.20}\] is unitary for all \(t\).
Proof. We first show that \(\hat{U}(t) = e^{-i\hat{H}t/\hbar}\) satisfies the Schrödinger equation: \[i\hbar \frac{d}{dt}\hat{U}(t) = i\hbar \cdot \frac{-i\hat{H}}{\hbar} e^{-i\hat{H}t/\hbar} = \hat{H}e^{-i\hat{H}t/\hbar} = \hat{H}\hat{U}(t) \tag{2.21}\]
Now we show \(\hat{U}(t)\) is unitary. The adjoint is: \[\hat{U}(t)^\dagger = \left(e^{-i\hat{H}t/\hbar}\right)^\dagger = e^{i\hat{H}^\dagger t/\hbar} = e^{i\hat{H}t/\hbar} \tag{2.22}\] where we used \(\hat{H}^\dagger = \hat{H}\) (self-adjointness). Therefore: \[\hat{U}(t)^\dagger\hat{U}(t) = e^{i\hat{H}t/\hbar}e^{-i\hat{H}t/\hbar} = e^{0} = \hat{I} \tag{2.23}\] and similarly \(\hat{U}(t)\hat{U}(t)^\dagger = \hat{I}\). ◻
Corollary 2.21 (Deterministic Evolution). The Schrödinger evolution is deterministic: given \(|\psi(0)\rangle\) and \(\hat{H}\), the state \(|\psi(t)\rangle\) is uniquely determined for all \(t\).
Proof. The Schrödinger equation is a first-order linear ODE. By the existence and uniqueness theorem for such equations, given any initial condition \(|\psi(0)\rangle\), there exists a unique solution \(|\psi(t)\rangle\) for all \(t\). ◻
Corollary 2.22 (Conservation of Probability). If \(|\psi(0)\rangle\) is normalized, then \(|\psi(t)\rangle\) is normalized for all \(t\).
Proof. Since \(\hat{U}(t)\) is unitary, it preserves norms: \[\langle \psi(t)|\psi(t)\rangle = \langle \hat{U}(t)\psi(0)|\hat{U}(t)\psi(0)\rangle = \langle \psi(0)|\hat{U}(t)^\dagger\hat{U}(t)|\psi(0)\rangle = \langle \psi(0)|\psi(0)\rangle = 1 \tag{2.24}\] ◻
Theorem 2.23 (Consistency of Born Probabilities). The probabilities given by the Born rule satisfy the axioms of probability theory:
\(P(a_i) \geq 0\) for all outcomes \(a_i\)
\(\sum_i P(a_i) = 1\) (probabilities sum to 1)
Proof. (1) Non-negativity: \(P(a_i) = |\langle a_i|\psi\rangle|^2 \geq 0\) since \(|z|^2 \geq 0\) for any complex number \(z\).
(2) Normalization: Let \(\{|a_i\rangle\}\) be an orthonormal basis of eigenstates of \(\hat{A}\). Then any \(|\psi\rangle\) can be expanded as: \[|\psi\rangle = \sum_i c_i|a_i\rangle, \quad \text{where } c_i = \langle a_i|\psi\rangle \tag{2.25}\] Since \(|\psi\rangle\) is normalized: \[1 = \langle \psi|\psi\rangle = \sum_{i,j} \overline{c_i}c_j\langle a_i|a_j\rangle = \sum_{i,j} \overline{c_i}c_j\delta_{ij} = \sum_i |c_i|^2 = \sum_i |\langle a_i|\psi\rangle|^2 = \sum_i P(a_i) \tag{2.26}\] ◻
Theorem 2.24 (Expectation Value). The expected value of observable \(\hat{A}\) in state \(|\psi\rangle\) is: \[\langle \hat{A} \rangle_\psi = \sum_i a_i P(a_i) = \langle \psi|\hat{A}|\psi\rangle \tag{2.27}\]
Proof. Let \(|\psi\rangle = \sum_i c_i|a_i\rangle\) where \(\{|a_i\rangle\}\) are orthonormal eigenstates of \(\hat{A}\) with eigenvalues \(a_i\). Then: \[\begin{align*} \langle \psi|\hat{A}|\psi\rangle &= \left(\sum_j \overline{c_j}\langle a_j|\right)\hat{A}\left(\sum_i c_i|a_i\rangle\right) \\ &= \sum_{i,j} \overline{c_j}c_i\langle a_j|\hat{A}|a_i\rangle \\ &= \sum_{i,j} \overline{c_j}c_i a_i\langle a_j|a_i\rangle \\ &= \sum_{i,j} \overline{c_j}c_i a_i\delta_{ij} \\ &= \sum_i |c_i|^2 a_i \\ &= \sum_i a_i P(a_i) \end{align*} \tag{2.28}\] ◻
We have now seen the fundamental tension at the heart of quantum mechanics:
Between measurements: The state evolves deterministically according to the Schrödinger equation. Given \(|\psi(0)\rangle\) and \(\hat{H}\), the state \(|\psi(t)\rangle\) is uniquely determined.
At measurements: The outcome is probabilistic according to the Born rule. The state “collapses” to an eigenstate in a random, non-deterministic manner.
Theorem 2.25 (The Dichotomy). Every event in a quantum universe is either:
Deterministically determined by the prior quantum state via Schrödinger evolution, or
Probabilistically determined by the Born rule upon measurement.
There is no third category.24
Corollary 2.26 (No Room for Free Will in Quantum Mechanics). Quantum mechanics provides no space for free will. The deterministic Schrödinger evolution excludes free will by making the future state a necessary consequence of the past state. The probabilistic Born rule excludes free will by making departures from determinism random rather than chosen.
Proof. By Chapter 1, Theorems 1.1 and 1.2, deterministic events and random events both fail to constitute exercises of free will. By Theorem 2.14, every quantum event falls into one of these two categories. Therefore, no quantum event constitutes an exercise of free will. ◻
Different interpretations of quantum mechanics handle the measurement problem differently, but none provides room for free will.25
The Copenhagen interpretation, developed by Bohr and Heisenberg, accepts both the deterministic Schrödinger evolution and the probabilistic collapse as fundamental.26
Proposition 2.27 (Copenhagen and Free Will). The Copenhagen interpretation excludes free will.
Proof. Between measurements, evolution is deterministic, excluding free will by Theorem 1.1. At measurements, outcomes are random, excluding free will by Theorem 1.2. There is no third regime where free will could operate. ◻
The Many-Worlds interpretation, proposed by Hugh Everett III (1957), denies that collapse ever occurs. The Schrödinger equation always applies; what appears as collapse is actually the branching of the universe into multiple branches, each realizing a different outcome.27
Proposition 2.28 (Many-Worlds and Free Will). The Many-Worlds interpretation excludes free will.
Proof. On the Many-Worlds view, the universal wave function evolves deterministically at all times—there is no collapse. By Theorem 1.1, determinism excludes free will. Therefore, Many-Worlds excludes free will.28 ◻
Bohmian mechanics, developed by David Bohm (1952) based on ideas of de Broglie, posits that particles have definite positions at all times, guided by the wave function according to the guidance equation: \[\frac{d\mathbf{Q}}{dt} = \frac{\hbar}{m}\text{Im}\left(\frac{\nabla\Psi}{\Psi}\right)\bigg|_{\mathbf{x}=\mathbf{Q}} \tag{2.29}\] where \(\mathbf{Q}\) is the particle position and \(\Psi\) is the wave function.29
Proposition 2.29 (Bohmian Mechanics and Free Will). Bohmian mechanics excludes free will.
Proof. In Bohmian mechanics, both the wave function (via Schrödinger’s equation) and the particle positions (via the guidance equation) evolve deterministically. Given the initial wave function \(\Psi_0\) and initial positions \(\mathbf{Q}_0\), all future states are uniquely determined. By Theorem 1.1, such determinism excludes free will.30 ◻
Theorem 2.30 (Universal Exclusion of Free Will by Quantum Mechanics). Every coherent interpretation of quantum mechanics excludes free will.
Proof. Any interpretation must either:
Accept genuine collapse with Born-rule randomness (like Copenhagen), which excludes free will by Theorem 1.2.
Deny collapse and accept purely deterministic evolution (like Many-Worlds or Bohmian mechanics), which excludes free will by Theorem 1.1.
Posit some combination, which excludes free will by Theorem 1.3.
These cases are exhaustive.31 ◻