Derivation of quantum theory from Feynman's rules

Feynman's formulation of quantum theory is remarkable in its combination of formal simplicity and computational power. However, as a formulation of the abstract structure of quantum theory, it is incomplete as it does not account for most of the fundamental mathematical structure of the standard von Neumann–Dirac formalism such as the unitary evolution of quantum states. In this paper, we show how to reconstruct the entirety of the finite-dimensional quantum formalism starting from Feynman's rules with the aid of a single physical postulate, the no-disturbance postulate. This postulate states that a particular class of measurements have no effect on the outcome probabilities of subsequent measurements performed. We also show how it is possible to derive both the amplitude rule for composite systems of distinguishable subsystems and Dirac's amplitude-action rule, each from a single elementary and natural assumption, by making use of the fact that these assumptions must be consistent with Feynman's rules.

Figure 1

Schematic representation of a Stern-Gerlach experiment performed on silver atoms. A silver atom from a source (an evaporator) is subject to a sequence of measurements, each of which yields one of two possible outcomes registered by nonabsorbing wire-loop detectors. Between measurements, the atoms interact with a uniform magnetic field. A run of the experiment yields outcomes $\ell ,m,n$ of the measurements $\mathbf{L},\mathbf{M},\mathbf{N}$ performed at times $t_1,t_2,t_3$, respectively. The probability distribution over the outcomes of $\mathbf{M}$ given an outcome of $\mathbf{L}$ is observed to be independent of any interactions the system had prior to $\mathbf{L}$, a property to which we refer as closure.

Figure 2

Composition of sequences belonging to two separate systems. In both examples, the first system undergoes measurements ${\mathbf{L}}_1,{\mathbf{M}}_1,{\mathbf{N}}_1$. The second system undergoes ${\mathbf{L}}_2,{\mathbf{M}}_2,{\mathbf{N}}_2$ in (a) and ${\mathbf{L}}_2,{\tilde{\mathbf{M}}}_2,{\mathbf{N}}_2$ in (b). For concreteness, the atomic measurements are assumed to have two possible outcomes, with $m_1,m_1^{\prime }$ labeling the possible outcomes of ${\mathbf{M}}_1$, and so on. In (a), the first system yields sequence $A=[{\ell }_1,m_1,n_1]$, and the second system yields $B=[{\ell }_2,m_2^{\prime },n_2]$, which are composed ($\times$) to yield sequence $C=[\ell ,m^{\prime },n]=[({\ell }_1:{\ell }_2),(m_1:m_2^{\prime }),(n_1:n_2)]$. In (b), the first system yields $A=[{\ell }_1,m_1,n_1]$, and the second system yields $B=[{\ell }_2,(m_2,m_2^{\prime }),n_2]$, which compose to yield $C=[\ell ,(m,m^{\prime }),n]=[({\ell }_1:{\ell }_2),((m_1:m_2),(m_1:m_2^{\prime })),(n_1:n_2)]$.

Figure 3

Feynman's rules for individual systems, expressed in operational terms. In each case, the sequence names are denoted $A,B,\cdots$, while their amplitudes are denoted $a,b,\cdots$. (a) Sum rule: $z(A\vee B)=z\left(A\right)+z\left(B\right).$ (b) Product rule: $z\left(A·B\right)=z\left(A\right)z\left(B\right)$. (c) Probability rule: Pr$(m,n\mid \ell =|a^2\left|\right).$

Figure 4

No-disturbance postulate. Left: A system undergoes measurement $\mathbf{L}$ at time $t$, followed by $\mathbf{N}$ at time $t^{\prime }$. For illustration, each measurement has two possible outcomes. The sequence of outcomes $[\ell ,n]$ has associated probability $Pr\left(n\right|\ell )$. Right: Trivial measurement $\tilde{\mathbf{M}}$, with single outcome $\tilde{m}$, occurs between $\mathbf{L}$ and $\mathbf{N}$. By the no-disturbance postulate, $\tilde{\mathbf{M}}$ has no effect on the probability of outcome $n$ given $\ell$.

Figure 5

Disturbance of repeatability. Left: A system undergoes measurement $\mathbf{L}$ at time $t$, and again immediately afterwards at $t^{\prime }$, with trivial measurement $\tilde{\mathbf{M}}$ in between. Since measurement $\mathbf{L}$ is a repeatable measurement, the no-disturbance postulate implies that it will yield the same outcome at $t^{\prime }$ as at $t$, even though $\tilde{\mathbf{M}}$ is present. Right: From a classical point of view, the occurrence of $\tilde{m}=(m,m^{\prime })$ implies that either outcome $m$ or $m^{\prime }$ occurred, but was not observed. The transition probabilities are as indicated, assuming that transition probabilities are symmetric. From these probabilities, it follows that repeatability is disturbed by $\tilde{\mathbf{M}}$ unless $\alpha$ is 0 or 1.

Figure 6

Left: A system undergoes measurement $\mathbf{L}$ at time $t$, followed by $\mathbf{N}$ at time $t^{\prime }$, yielding sequence $[{\ell }^{\left(1\right)},n^{\left(1\right)}]$. Middle: If trivial measurement $\tilde{\mathbf{M}}$ occurs immediately prior to $\mathbf{N}$, then, by the no-disturbance postulate, it has no effect on the outcome probabilities of $\mathbf{N}$. Hence, the probability $Pr\left(n^{\left(1\right)}\right|{\ell }^{\left(1\right)})={\left|{\tilde{v}}_1\right|}^2$. Right: From the amplitude sum rule, ${\tilde{v}}_1=v_1{T}_{11}+v_2{T}_{12}={\left(\mathsf{\text{T}}\mathsf{\text{v}}\right)}_1.$

Figure 7

Illustration of the cross-multiplicativity property. The composite sequence $[({\ell }_1:{\ell }_2),(m_1:m_2),(n_1:n_2)]$ can be obtained by combining the sequences $A=[{\ell }_1,m_1],B=[m_1,n_1],C=[{\ell }_2,m_2]$, and $D=[m_2,n_2]$ in two different ways, as shown, yielding the cross-multiplicativity relation $\left(A·B\right)\times \left(C·D\right)=(A\times C)·(B\times D)$, which is Eq. (30).

Figure 8

Illustration of left-distributivity of $\times$ over $\vee$. The composite sequence $[({\ell }_1,{\ell }_2),(m_1:(m_2,m_2^{\prime })),(n_1,n_2)]$ can be obtained by combining the sequences $A=[{\ell }_1,m_1,n_1]$, $B=[{\ell }_2,m_2,n_2]$, and $C=[{\ell }_2,m_2^{\prime },n_2]$ in two different ways, as shown, yielding the relation $A\times (B\vee C)=(A\times B)\vee (A\times C)$, which is Eq. (31).

Figure 9

The amplitude of the path $C=A·B$ can be obtained from the classical actions $S_A,S_B$ of paths $A,B$ in two different ways: (i) obtain the action $S_C=S_A+S_B$, whose corresponding amplitude is $f(S_A+S_B)$; and (ii) use $f$ to obtain the amplitudes $f\left(A\right),f\left(B\right)$ and then compose these to obtain amplitude $f\left(S_A\right)f\left(S_B\right)$. Hence, $f(S_A+S_B)=f\left(S_A\right)f\left(S_B\right)$.