where the superscript \(\alpha_m\) labels different possible projectors at time \(t_m\), and \(\alpha = (\alpha_1,\alpha_2,\ldots\alpha_f)\) is a composite label for the entire history. (Since the square of a projector is the same as the projector, the superscript position is not needed for exponents, and it is convenient to reserve the subscript to indicate the time.) If for each time \(t_m\) it is the case that \(\sum_{\alpha_m} P_m^{\alpha_m} = I_m\) it follows that \(\sum_\alpha Y^\alpha = \breve I\), the identity operator on \(\breve\HC\); thus the \(\{Y^\alpha\}\) form a projective decomposition of the identity \(\breve I\), a quantum sample space or family of histories, one and only one of which will actually occur in any given experimental run. The physical interpretation of \eqref{eqn13} is that the property \(P_1^{\alpha_1}\) occurs or is true at time \(t_1\), \(P_2^{\alpha_2}\) at time \(t_2\), and so forth. Indeed, since we live in a quantum world, the “classical” coin-tossing history \(THT\) can be written in the form \eqref{eqn13} using appropriate projectors to represent \(T\) and \(H\). As will be discussed later in section 6, the histories approach employs the same mathematical procedures for both microscopic and macroscopic physics, consistent with the belief of most physicists that quantum theory applies to phenomena of any size.