Weak Value, Quasiprobability and Bohmian Mechanics

Abstract

We clarify the significance of quasiprobability (QP) in quantum mechanics that is relevant in describing physical quantities associated with a transition process. Our basic quantity is Aharonov’s weak value, from which the QP can be defined up to a certain ambiguity parameterized by a complex number. Unlike the conventional probability, the QP allows us to treat two noncommuting observables consistently, and this is utilized to embed the QP in Bohmian mechanics such that its equivalence to quantum mechanics becomes more transparent. We also show that, with the help of the QP, Bohmian mechanics can be recognized as an ontological model with a certain type of contextuality.

Appendix: Some Useful Formulas for the \(\circ _{\alpha }\)-Product

The $\circ _{\alpha }$-product (16) for two operators X and Y on $\mathscr {H}$ is defined by

$$\begin{aligned} X\circ _{\alpha }Y:=\alpha XY+\left( 1-\alpha \right) YX, \end{aligned}$$

(67)

with $\alpha \in \mathbb {C}$. For $\alpha =1$ and $\alpha =0$, the $\circ _{\alpha }$-product becomes the usual operator product,

$$\begin{aligned} X\circ _{\alpha =1}Y=XY, \qquad X\circ _{\alpha =0}Y=YX, \end{aligned}$$

(68)

whereas for $\alpha =1/2$ it reduces to the Jordan product [33],

$$\begin{aligned} X\circ _{\alpha =\frac{1}{2}}Y=\frac{1}{2}\left( XY+YX\right) . \end{aligned}$$

(69)

If we put $\alpha = s + it$ with real s, t, we have

$$\begin{aligned} X\circ _{\alpha =s+it}Y = s XY+\left( 1-s\right) YX+it \left[ X,Y\right] = X\circ _s Y+it\left[ X,Y\right] , \end{aligned}$$

(70)

where $[X,Y]=XY-YX$ is the commutator. In particular, for $\alpha =\frac{1}{2}\left( 1-i\right)$ we find

$$\begin{aligned} X\circ _{\alpha =\frac{1}{2}\left( 1-i\right) }Y=\frac{1}{2}\{X,Y\}+\frac{1}{2i}[X,Y], \end{aligned}$$

(71)

where $\{X,Y\}=XY+YX$ is the anti-commutator. In addition, the $\circ _{\alpha }$-product has the following properties:

$$\begin{aligned} I\circ _{\alpha }X= & {} X\circ _{\alpha }I=X, \end{aligned}$$

(72)

$$\begin{aligned} X\circ _{\alpha }X= & {} X^{2},\end{aligned}$$

(73)

$$\begin{aligned} X\circ _{\alpha }Y-Y\circ _{\alpha }X= & {} \left( 2\alpha -1\right) [X,Y], \end{aligned}$$

(74)

$$\begin{aligned} X\circ _{\alpha }Y+Y\circ _{\alpha }X= & {} XY+YX, \end{aligned}$$

(75)

$$\begin{aligned} {} [X , Y] = 0\,\Rightarrow & {} \, X \circ _{\alpha } Y=XY=YX, \end{aligned}$$

(76)

with I being the identity operator on $\mathscr {H}$.