**Complementarity Principle**

Niels Bohr introduced and explained his concept of "complementarity" in
his famous 1927 Como Lecture (reproduced in Bohr 1934). He recognized
the need for the mathematical formalism of quantum mechanics to be
imbedded a rationally coherent conceptual framework if it were to serve
as the core of an acceptable scientific theory. Yet the applications of
the formalism were based upon the integration of two logically
incompatible conceptual structures, the mathematical formalisms of
classical and quantum physics. The applications that we normally make of
quantum theory involve three physical systems: (1), the system being
examined; (2), the measuring devices by means of which we probe its
properties; and (3), our own physical bodies. All three systems are
composed of atoms, and hence must be describable in terms of the
mathematical concepts of quantum theory. Yet our observations are
described in terms of the contents of our sense experiences, which, for
the phenomena under consideration, are described in terms of the
concepts of classical physics.

Classical physics postulates that, at each instant of time, each
elementary particle is located at some definite point in space, and has
a definite velocity, and hence also a definite momentum. On the other
hand, in quantum mechanics an elementary particle is represented by a
distribution of possibilities, where the distributions in position and
in momentum are related by Fourier transformation. This entails that
localization at a point in position space demands a complete lack of
localization in momentum space, and vice versa. Bohr associates
"causation" with the law of conservation of momentum and energy, and
hence is able to say that:

> The very nature of quantum theory thus forces us to regard the claim
> of space-time co-ordination and the claim of causality, the union of
> which characterizes the classical theories, as complementary but
> exclusive features of the description, symbolizing the idealization of
> observation and definition respectively. (Bohr 1934, p. 54)

Bohr explains that:

> The quantum theory is characterized by the acknowledgement of a
> fundamental limitation in the classical physical ideas when applied to
> atomic phenomena. ...its essence may be expressed in the so-called
> quantum postulate, which attributes to any atomic process an essential
> discontinuity, or rather individuality, completely foreign to
> classical theories and symbolized by Planck's quantum of action....the
> quantum postulate implies that any observation of atomic phenomena
> will involve an interaction with the agency of observation not to be
> neglected. Accordingly, an independent reality in the ordinary
> physical sense can neither be ascribed to the phenomena nor to the
> agencies of observation. After all, the concept of observation is in
> so far arbitrary as it depends upon which objects are included in the
> system to be observed. Ultimately, every observation can, of course,
> be reduced to our sense perceptions." (Bohr 1934, p. 53)

These passages gives a glimpse of the range and complexity of the ideas
that Bohr wants to integrate into his rationally coherent foundation for
the application and use of quantum theory.

The elaboration that he provides in the remainder of the Como lecture is
lengthy, but its essence is summarized and updated in his 1958 paper
"Quantum physics and Philosophy: Causality and Complementarity", in
which he says:

> Within the scope of classical physics, all characteristic properties
> of a given object can in principle be ascertained by a single
> experimental arrangement, although in practice various arrangements
> are often convenient for the study of different aspects of the
> phenomena. In fact, data obtained in such a way simply supplement each
> other and can be combined into a consistent picture of the behaviour
> of the object under investigation. In quantum mechanics, however,
> evidence about atomic objects obtained by different experimental
> arrangements exhibits a novel kind of complementary relationship.
> Indeed, it must be recognized that such evidence which appears
> contradictory when combination into a single picture is attempted,
> exhaust all conceivable knowledge about the object. Far from
> restricting our efforts to put questions to nature in the form of
> experiments, the notion of *complementarity* simply characterizes the
> answers we can receive by such inquiry, whenever the interaction
> between the measuring instruments and the objects form an integral
> part of the phenomena.
>
> (Bohr 1962, p.4)

Compactly stated, the essential idea here is that in quantum theory the
information provided by different experimental procedures that in
principle cannot, because of the physical character of the needed
apparatus, be performed simultaneously, cannot be represented by any
mathematically allowed quantum state of the system being examined. The
elements of information obtainable from incompatible measurements are
said to be complementary: taken together exhaust the information
obtainable about the state. On the other hand, any preparation protocol
that is maximally complete, in the sense that all the procedures are
mutually compatible and are such that no further procedure can add any
more information, can be represented by a quantum state, and that state
represents in a mathematical form all the conceivable knowledge about
the object that experiments can reveal to us.

As regards the closely connected issue of causality, Bohr says:

> In the treatment of atomic problems, actual calculations are most
> conveniently carried out with the help of a Schroedinger state
> function, from which the statistical laws governing observations
> obtainable under specified conditions can be deduced by definite
> mathematical operations. It must be recognized, however, that we are
> dealing here with a purely symbolic procedure, the unambiguous
> physical interpretation of which in the last resort requires reference
> to the complete experimental arrangement. (Bohr 1963, p. 5)

This relegation of the Schroedinger state function, which gives the
space-time representation of the atomic substrate of all systems, to a
purely symbolic status, might seem to be denigrating this Schroedinger
representation of the state relative to others. But the point is rather
that it puts the Schroedinger space-time representation on a par with
the others:

> In fact, wave mechanics, just as the matrix theory, represents on this
> view a symbolic transcription of the problem of motion of classical
> mechanics adapted to the requirements of quantum theory and only to be
> interpreted by an explicit use of the quantum postulate. (Bohr 1934,
> p.75)

All of this must be understood within the basic pragmatic premise of
Bohr's approach:

> In our description of nature the purpose is not to disclose the real
> essence of phenomena but only to track down as far as possible
> relations between the multifold aspects of our experience. (Bohr 1934,
> p. 18)

**References**

Bohr, N. (1934): *Atomic theory and the description of nature*.
Cambridge University Press, Cambridge UK.

Bohr, N. (1963): *Essays 1958-1962 on Atomic Physics and Human
Knowledge.* Wiley, New York.