**Quantum Zeno Effect in the Brain^1^**

The effects of environmentally induced decoherence are most conveniently
expressed in terms of the density matrix D. The classic paper of Joos
and Zeh^2^ shows that the effect upon a system of its interaction with
its environment is generally to damp out rapidly the off-diagonal
elements of its coordinate-space density matrix D.

The quantum Zeno effect is a consequence of repeatedly posing, in
sufficiently rapid succession, the *same* measurement question. In the
case of a question with a 'Yes' or 'No' answer, the outcome 'Yes' will
be associated with a particular (projection) operator P, satisfying
PP=P, and the act of posing the question\-\--von Neumann's famous
Process1\-\-- will then be represented by replacing D by PDP +
(1-P)D(1-P). If the answer to the question is 'Yes', the density matrix
is reduced to PDP/Trace PDP. Then the fundamental interpretive rule of
quantum mechanics (See von Neumann^3^) asserts that the probability that
the outcome of the succeeding measurement question will be 'No' is

Trace (1-P) exp --iHt PDP exp iHt (1-P)/Trace PDP. Here H is the
Hamiltonian and t is the time difference between the two measurements.
Introducing for small t the power series expansion of the exponentials,
and using (1-P)P=P(1-P)=0, one finds that the term lowest order in t is
of order t squared.

This result implies that as the timing of the questions becomes rapid,
on the scale of the important time scales in PDP, the evolution of the
system to states where the property specified by 'Yes' fails to hold
will become increasingly suppressed. This conclusion is independent of
whether the off-diagonal elements of D are large or small. The
decoherence-inducing term in the master equation does not affect this
conclusion.

In the neural context the 'Yes' states are those in which the neural
correlate of the conscious intention is active. According to Koch and
Hepp, this activity extends over a macroscopic portion of the brain:
this quantum effect is intrinsically macroscopic. The timings of the
repetitious acts of measurement are not specified by any known laws:
according to orthodox quantum ideas, the choices of the measurement
acts, and of their timings, are both, in Bohr's words, "free choices" on
the part of the human participant: they are constrained by no known law,
statistical or otherwise.

This quantum Zeno effect can provide a physics-based causal explanation
of the claim of William James^4^ that

> No object can *catch* our attention except by the neural machinery.
> But the *amount* of the attention which an object receives after it
> has caught our attention is another question. It often takes effort to
> keep the mind upon it. We feel that we can make more or less of the
> effort as we choose. If this feeling be not deceptive, if our effort
> be a spiritual force, and an indeterminate one, then of course it
> contributes coequally with the cerebral conditions to the result.
> Though it *introduce* no new idea, it will deepen and prolong the stay
> in consciousness of innumerable ideas which else would fade
> more quickly away.
>
> Everywhere, then, the function of effort is the same: to keep
> affirming and adopting the thought which, if left to itself, would
> slip away.

1.  Misra, B. & Sudarshan, E. The Zeno's Paradox in Quantum Theory. J.
    Math. Phys. 18, 756-763 (1977).

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1.  Joos, E. & Zeh, H. The Emergence of Classical Properties Through
    Interaction with the Evironment. *Z. Phys.* B59, 223-243 (1985).

2.  Von Neumann, J. *Mathematical Foundations of Quantum Mechanics*
    (Princeton University Press, Princeton, 1955).

3.  James, W. Psychology: The Briefer Course in *William James: Writings
    1879-1899.* pp.227, 421. (New Library of America, New York, 1992)