# Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

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## Abstract

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## 1. Introduction

- (a)
- Amplitude Sum Rule: If a system classically can take $n>1$ possible paths from ${E}_{i}$ to ${E}_{f}$, but the experimental apparatus does not permit one to determine which path was taken, then the total amplitude, z, for the transition from ${E}_{i}$ to ${E}_{f}$ is given by the sum of the amplitudes, ${z}_{k}$, associated with these paths, so that $z={\sum}_{k=1}^{n}{z}_{k}$;
- (b)
- Amplitude Product Rule: If the transition from ${E}_{i}$ to ${E}_{f}$ takes place via intermediate event ${E}_{m}$, the total amplitude, u, is given by the product of the amplitudes, ${u}^{\prime}$ and ${u}^{\u2033}$, for the transitions ${E}_{i}\to {E}_{m}$ and ${E}_{m}\to {E}_{f}$, respectively, so that $u={u}^{\prime}{u}^{\u2033}$; and
- (c)
- Probability-Amplitude Rule: The probability, ${p}_{{E}_{i}\to {E}_{f}}$, of the transition from ${E}_{i}$ to ${E}_{f}$ is equal to the modulus-squared of the total amplitude, z, for the transition, so that ${p}_{{E}_{i}\to {E}_{f}}={\left|z\right|}^{2}$.

- Operational Framework: First, we establish a fully operational framework in which to describe measurements performed upon physical systems. The framework allows the results of an experiment to be described in purely operational terms by simply stating which sequence of measurements was performed and what were their results. In particular, any metaphysical speculations or physical pictures about how a system behaves between measurements (such as imagining “classical paths” of a “particle” between initial and final position measurements as envisaged in Feynman’s rules) is eschewed.
- Experimental Logic: Second, we identify an experimental logic in which parallel and series operators can be used to combine together sequences of measurement outcomes obtained in experiments. These measurement sequences list the outcomes obtained when a sequence of measurements are performed on a physical system. The action of applying the logical parallel and series operators allows us to formally relate the results of different experiments. The logic itself is characterized by five symmetries that are induced by the operational definition of these operators.
- Process Calculus: Third, we represent these measurement sequences with pairs of real numbers, this choice of representation being inspired by the principle of complementarity articulated by Bohr [12]. This representation induces a pair-valued calculus characterized by a set of functional equations, which are then solved to yield the possible forms of the two pair operators which correspond to the parallel and series sequence operators.
- Connection with Probability Theory: Fourth, and finally, we associate a logical proposition with each measurement sequence, and postulate that the pair associated with each sequence determines the probability of this proposition. We further require that (a) the calculus be consistent with probability theory when applied to series-combined sequences, and (b) when applied to parallel-combined sequences, the maximum and minimum values of the probabilistic predictions of the calculus are placed symmetrically about what one would predict using probability theory on the assumption that these sequences are probabilistically independent (an assumption which follows from classical physics). The resulting calculus—which we refer to as the process calculus—coincides with Feynman’s rules of quantum theory.

## 2. Symmetries in Probability Theory

**C**= “it is cloudy” implies the statement

**R**= “it is raining”, $Pr\left(\mathbf{R}\phantom{\rule{0.166667em}{0ex}}\right|\phantom{\rule{0.166667em}{0ex}}\mathbf{C})$, should be equal to the degree to which the statement (

**C**,

**E**) = “it is cloudy, and eggplants are purple” implies the statement (

**R**,

**E**) = “it is raining, and eggplants are purple”, denoted $Pr\left((\mathbf{R},\mathbf{E})\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right(\mathbf{C},\mathbf{E})\right)$.

## 3. Feynman’s Rules of Quantum Theory

**A**= “Electron is emitted from A at time t_{1}”**B**= “Electron passes through the union of the space occupied by slits B_{1}and B_{2}at time t_{2}”**C**= “Electron is detected at given cell at C at time t_{3}”

**B**_{1}= “Electron passes through slit B_{1}at time t_{2}”**B**_{2}= “Electron passes through slit B_{2}at time t_{2}”

## 4. Derivation of Feynman’s Rules

#### 4.1. Operational Experimental Framework

#### 4.2. Sequence Combination Operators

**process calculus**—that is capable of establishing a relation between the probabilities observed in experimental set-ups such as those in Figure 2 and Figure 3. For example, in a run of the first experiment, one might observe the sequences $A=[1,1,1]$ or $B=[1,2,1]$, while, in a run of the second experiment, one might observe $C=[1,(1,2),2]$. The calculus should provide a relationship between the probabilities $P\left(A\right)$, $P\left(B\right)$ and $P\left(C\right)$ associated with these sequences.

#### 4.3. Pair Representation of Sequences

#### 4.4. Probabilities, and the Probability Product Equation

- Case C1: $p\left(\mathbf{a}\right)={\left({a}_{1}^{2}+{a}_{2}^{2}\right)}^{\alpha /2}$;
- Case C2: $p\left(\mathbf{a}\right)=|{a}_{1}{|}^{\alpha}{e}^{\beta {a}_{2}/{a}_{1}}$;
- Case C3: $p\left(\mathbf{a}\right)=|{a}_{1}{|}^{\alpha}{\left|{a}_{2}\right|}^{\beta}$;
- Case N1: $p\left(\mathbf{a}\right)=|{a}_{1}{|}^{\alpha}$;
- Case N2: $p\left(\mathbf{a}\right)=|{a}_{1}{|}^{\alpha}$;

#### 4.5. Pair Symmetry

#### 4.6. Independent Parallel Processes

**Additivity Condition:**For any given probabilities ${p}_{1}$ and ${p}_{2}$ for which ${p}_{1}+{p}_{2}\le 1$, there exist pairs $\mathbf{a}$ and $\mathbf{b}$ satisfying $p\left(\mathbf{a}\right)={p}_{1},p\left(\mathbf{b}\right)={p}_{2}$ such that Equation (76) (which we shall henceforth refer to as the additivity equation) holds true whenever $p(\mathbf{a}+\mathbf{b})\le 1$.

#### 4.7. Case (C1), with $p\left(\mathbf{x}\right)={({x}_{1}^{2}+{x}_{2}^{2})}^{\alpha /2}$

## 4.8. Symmetric Bias

**Symmetric Bias Condition [25]:**For any given probabilities ${p}_{1}$ and ${p}_{2}$ for which ${p}_{1}+{p}_{2}\le 1$, there exist pairs $\mathbf{a}$ and $\mathbf{b}$ satisfying $p\left(\mathbf{a}\right)={p}_{1},p\left(\mathbf{b}\right)={p}_{2}$ such that ${\beta}_{+}={\beta}_{-}$ holds true whenever $p(\mathbf{a}+\mathbf{b})\le 1$.

## 5. Summary

- As shown on the right hand side of the diagram, statements $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are all atomic; in particular, $\mathbf{C}$ cannot be obtained from $\mathbf{A}$ and $\mathbf{B}$ by means of any Boolean logical operations.
- In probability theory unfettered by additional constraints, the probabilities of the atomic statements $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ would be freely assignable (see Section 2). However, additional constraints do exist, as a result of which these probabilities are not freely assignable. More precisely, (i) due to the amplitude sum rule operative in the pair space, the pair representing sequence C is determined by the pairs representing A and B and (ii) due to the postulated connection between the sequence space and the statement space, the probability of $\mathbf{C}$ is determined by the pairs representing sequences A and B. That is, once ${z}_{1}$ and ${z}_{2}$ are fixed, the probabilities of not only propositions $\mathbf{A}$, $\mathbf{B}$, but also proposition $\mathbf{C}$, are determined.
- In the statement space, the probability of proposition $\mathbf{C}$ is not independent of the probabilities of propositions $\mathbf{A}$, $\mathbf{B}$, but, on the other hand, is not determined by them either. The lee-way that exists in the probability of $\mathbf{C}$ even after the probabilities of $\mathbf{A}$ and $\mathbf{B}$ have been fixed arises because these three probabilities are determined through three independent degrees of freedom, namely $|{z}_{1}|$, $|{z}_{2}|$, and $arg({z}_{1}/{z}_{2})$, in pair space.
- In the statement space, one can construct the statement $\mathbf{A}\vee \mathbf{B}$ from $\mathbf{A}$ and $\mathbf{B}$ using the Boolean OR operation. The probability of $\mathbf{A}\vee \mathbf{B}$ is determined by the sum rule of probability theory that is operative in the probability space (which, in turn, results from the associative symmetry of the logical OR operation). In particular, statement $\mathbf{A}\vee \mathbf{B}$ is not the same as $\mathbf{C}$.
- The application of probability theory alone does not predict any quantitative relation between the probabilities of $\mathbf{A}\vee \mathbf{B}$ and $\mathbf{C}$. If one adds the appropriate assumption from classical physics, then these two propositions can be equated, which implies that the probability of $\mathbf{C}$ is given by the probability of $\mathbf{A}\vee \mathbf{B}$. Feynman’s rules posit an alternative set of assumptions (which we have explicitly identified in the process of deriving Feynman’s rules), which lead to the assignment of different probabilities to these two propositions.

#### 5.1. Real Quantum Theory

## 6. Conclusions

## Acknowledgements

## References and Notes

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**Figure 1.**Sketch of the double-slit experiment. On the left, a heated electrical filament serves as an electron source. Electrons emerging from the filament are collimated, and are detected as they pass through a wire-loop detector. The electrons then encounter a screen, B, containing two slits, and the electrons that pass B are registered by another wire-loop detector. Finally, the electrons that pass through B will be detected on the screen on the right-hand side.

**Figure 2.**An experimental set-up consisting of three successive measurements, each of which has two possible outcomes. In a particular run of the experiment, the measurement outcome sequence $[1,1,1]$ is obtained.

**Figure 3.**An experimental set-up consisting of three measurements in which the second measurement is coarse-grained. In a particular run of the experiment, the sequence $[1,(1,2),1]$ is obtained.

**Figure 4.**Illustration of the overall logical structure of the process calculus. On the top left is the space of sequences. To each sequence, A, corresponds a conditional statement, $\mathbf{A}$ (top right). Each sequence is represented by a pair, $\mathbf{a}$ (bottom left), while each conditional statement is represented by a probability, $Pr\left(\mathbf{A}\right)$ (bottom right). The link between these representations, $Pr\left(\mathbf{A}\right)=p\left(\mathbf{a}\right)$, is given on the bottom right.

**Figure 5.**For $p\left(\mathbf{a}\right)=p\left(\mathbf{b}\right)=1/8$, this graph shows, as a function of $\alpha $, (i) the extreme values of $p(\mathbf{a}+\mathbf{b})$, and (ii) the value of $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)=1/4$. For $\alpha <1$, the maximum of $p(\mathbf{a}+\mathbf{b})$ is less than $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)$.

**Figure 6.**For $p\left(\mathbf{a}\right)=p\left(\mathbf{b}\right)=1/8$, this graph shows, as a function of $\alpha $, (i) the extreme values of $p(\mathbf{a}+\mathbf{b})$, and (ii) the value of $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)=1/4$. For $\alpha <1/2$, the maximum of $p(\mathbf{a}+\mathbf{b})$ is less than $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)$.

**Figure 7.**Graphs (a) and (b) show, as a function of $\alpha $ for the indicated values of $p\left(\mathbf{a}\right)$ and $p\left(\mathbf{b}\right)$, (i) the extreme values of $p(\mathbf{a}+\mathbf{b})$, (ii) the average of these extrema, and (iii) the value of $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)$. In both cases, the average of the extrema coincides with the value of $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)$ only when $\alpha =2$.

**Figure 8.**Graphs (a) and (b) show, as a function of $\alpha $ for the indicated values of $p\left(\mathbf{a}\right)$ and $p\left(\mathbf{b}\right)$, (i) the extreme values of $p(\mathbf{a}+\mathbf{b})$, (ii) the average of these extrema, and (iii) the value of $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)$. The average of the extrema coincides with the value of $p\left(\mathbf{a}\right)+p\left(\mathbf{b}\right)$ at different values of $\alpha $ in the two graphs.

**Figure 9.**A diagram illustrating the connection between the space of measurement sequences and the space of statements. On the left hand side, the sequences A and B are combined together in parallel to generate sequence $C=A\vee B$. If amplitudes ${z}_{1}$ and ${z}_{2}$ represent sequences A and B, respectively, then, by the amplitude sum rule, amplitude ${z}_{1}+{z}_{2}$ represents sequence C. On the right hand side, corresponding to the sequences A, B, and C are the atomic statements $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$, with probabilities $|{z}_{1}{|}^{2}$, $|{z}_{2}{|}^{2}$, and $|{z}_{1}+{z}_{2}{|}^{2}$, respectively. Note that the probability associated with $\mathbf{C}$ is not freely assignable due to the postulated connection between the two spaces. Also shown is the statement $\mathbf{A}\vee \mathbf{B}$, which is distinct from $\mathbf{C}$, and which has probability $|{z}_{1}{|}^{2}+{\left|{z}_{2}\right|}^{2}$ determined by the sum rule of probability theory.

Unary Operation | |

Complementation | $\mathrm{NOT}\phantom{\rule{3.33333pt}{0ex}}\equiv \phantom{\rule{3.33333pt}{0ex}}\neg $ |

Complementation 1 | $\mathbf{A}\wedge \neg \mathbf{A}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\perp $ |

Complementation 2 | $\mathbf{A}\vee \neg \mathbf{A}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\top $ |

Idempotency | $\mathbf{A}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\neg \neg \mathbf{A}$ |

Binary Operations | |

Disjunction | $\mathrm{OR}\phantom{\rule{3.33333pt}{0ex}}\equiv \phantom{\rule{3.33333pt}{0ex}}\vee $ |

Conjunction | $\mathrm{AND}\phantom{\rule{3.33333pt}{0ex}}\equiv \phantom{\rule{3.33333pt}{0ex}}\wedge $ |

Idempotency | $\mathbf{A}\vee \mathbf{A}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathbf{A}$ |

$\mathbf{A}\wedge \mathbf{A}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathbf{A}$ | |

Commutativity | $\mathbf{A}\vee \mathbf{B}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathbf{B}\vee \mathbf{A}$ |

$\mathbf{A}\wedge \mathbf{B}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathbf{B}\wedge \mathbf{A}$ | |

Associativity | $\mathbf{A}\vee (\mathbf{B}\vee \mathbf{C})\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}(\mathbf{A}\vee \mathbf{B})\vee \mathbf{C}$ |

$\mathbf{A}\wedge (\mathbf{B}\wedge \mathbf{C})\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}(\mathbf{A}\wedge \mathbf{B})\wedge \mathbf{C}$ | |

Absorption | $\mathbf{A}\vee (\mathbf{A}\wedge \mathbf{B})\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathbf{A}\wedge (\mathbf{A}\vee \mathbf{B})\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathbf{A}$ |

Distributivity | $\mathbf{A}\wedge (\mathbf{B}\vee \mathbf{C})\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}(\mathbf{A}\wedge \mathbf{B})\vee (\mathbf{A}\wedge \mathbf{C})$ |

$\mathbf{A}\vee (\mathbf{B}\wedge \mathbf{C})\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}(\mathbf{A}\vee \mathbf{B})\wedge (\mathbf{A}\vee \mathbf{C})$ | |

De Morgan 1 | $\neg \mathbf{A}\wedge \neg \mathbf{B}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\neg (\mathbf{A}\vee \mathbf{B})$ |

De Morgan 2 | $\neg \mathbf{A}\vee \neg \mathbf{B}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\neg (\mathbf{A}\wedge \mathbf{B})$ |

Consistency | |

$\mathbf{A}\to \mathbf{B}\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}\mathbf{A}\wedge \mathbf{B}=\mathbf{A}\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}\mathbf{A}\vee \mathbf{B}=\mathbf{B}$ |

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Goyal, P.; Knuth, K.H. Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry. *Symmetry* **2011**, *3*, 171-206.
https://doi.org/10.3390/sym3020171

**AMA Style**

Goyal P, Knuth KH. Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry. *Symmetry*. 2011; 3(2):171-206.
https://doi.org/10.3390/sym3020171

**Chicago/Turabian Style**

Goyal, Philip, and Kevin H. Knuth. 2011. "Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry" *Symmetry* 3, no. 2: 171-206.
https://doi.org/10.3390/sym3020171