This paper reviews arguments for the necessity of complex numbers in any quantum theory.

1. Complex Numbers in Quantum Theory
PHY4910 – Spring 2010 – L. White
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Abstract.
This paper reviews arguments by Saxon
1
and Stueckelberg
2
for the necessity of complex numbers in any
quantum theory. It concludes that the twoarguments are complementary, and the use of complex
numbersin quantum theory isessential. As there is ongoing and promising research on the use of
quaternions as a construct for quantum theory
3
, it would be more correct to say that quantum theory is
impossible without the imaginary unit i. (Quaternions may be regarded as an extension of the complex
numbers.) It is which is woven into the fabric of quantum descriptions, in a way not seen in other
theories. For simplicity, this paper refers only to complex numbers, but it should be remembered that
these may beinterpreted as a limiting case of quaternions.
Introduction.
Complex numbers permeate quantum physics. Whereas in classical physics they are used as a powerful
notational shorthand, they are unavoidable in quantum theory. Schrödinger was following established practice
when he introduced a complex phase factor into his wave equation for the quantum state.
Ψ(x,t) = Aei(k.x – ωt)
(1)
But as quantum theory developed and matured, it became clear that complex numbers playeda critical role at
the heart of the theory. This raised the question, “Can you have a quantum theory without complex numbers?”
The answer from both Saxon and Stueckelberg was an emphatic “no”. This sets quantum theory apart; so what
is it about quantum theory that requires i?
Saxon.
Saxon‟s argument is concise and may be summarized as follows. All points in space must be physically
equivalent – the choice of origin should be irrelevant. If, for example, we shift the origin left by an arbitrary
distance b, the function Ψ from equation (1) is multiplied by phase factor eipb/ħ
. This leaves the wave function in
the same general form Ψ‟
(x,t) = Aei(k’.x’ – ωt)
, with no physically significant reference to the origin. Unlike the case
of a classical wave, the phase factor eipb/ħ
is physically undetectable; therefore, under an arbitrary
transformation, Ψ reduces to a multiple of itself. This essential aspect of the state function can only be
implemented in a complex framework. So what began as a notational convenience has taken on the character
of something more fundamental.
When determining probability density, for example, we multiply Ψ by its complex conjugate, giving |Ψ|2
. The
result, being a measure of probability, is – as you would expect – a dimensionless number, without phase or
direction. So while the state function itself is described by a wave equation that includes a complex phase
factor, there is no phase involved in its physical manifestation |Ψ|2
. (We do not observe electrons, for example,
as having an absolute phase, although their spatial distribution about the nucleus depends, albeit indirectly, on
Ψ.) This is a critical difference from classical physics, and for good reason: in classical physics, the wave
equation describes the physical phenomenon (e.g. an electromagnetic wave); in quantum theory, the wave
equation describes nothing directly; it serves only as a predictor of physical phenomena when „squared‟
through multiplication by its complex conjugate.

2. Complex Numbers in Quantum Theory
PHY4910 – Spring 2010 – L. White
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Stueckelberg.
Stueckelberg‟s argument is more technical than Saxon‟s. Finkelstein, Jauch, and Speiser4
had already shown
that the only viable Hilbert spaces which could underpin quantum theory were Real Hilbert Space (RHS),
Quaternion*
. Hilbert Space (QHS), and Complex Hilbert Space (CHS).Assuming nothing that wasn‟t obvious,
axiomatic, or trivial, Stueckelbergwas able to rule out RHS as a mathematical structure for quantum theory. He
starts with a generic classical definition of error operators and arrives at two possible definitions of mean
square error. These allow him to calculate the uncertainty in simultaneous measurements of two observables.
One of these gives the trivial result
<∆F2
>ψ<∆G2
>ψ ≥ 0 (2)
In other words, the uncertainty is greater than zero, unless it isn‟t. The second mean square calculation leads
to the result
<∆F2
>ψ<∆G2
>ψ ≥ 1/4 <J[F,G]>2
ψ (3)
This cannot be manipulated to give the canonical commutation relation
[a,b] = iħ (4)
or the Heisenberg Uncertainty Principle
ΔxΔp ≥ ħ/2 (5)
So the assumption of a real Hilbert space takes us to a dead end which cannot be reconciled with some of the
most basic aspects of quantum theory.
Stueckelberg‟s proof is laid out in detail in Appendix A (below).
Complementary and Related Arguments, and the Wheeler Question.
Saxon‟s argument centers on arbitrary transformations of the state function Ψ; Stueckelberg shows that Real
Hilbert spaces lead to trivial or unusable uncertainty principles. Stueckelberg‟s argument is more austere but
both arguments are compelling.They do not appear to be equivalent (in the sense, for example, that
Schrödinger‟s wave formulation turned out to be equivalent to Heisenberg‟s matrix mechanical
approach5
).They clearly complement each other. Saxon cautions us that, if we accept that the de Broglie wave
is the physical embodiment of the state function Ψ, we are also implicitly accepting the hard-wiring of into
the core of quantum physics. Stueckelberg demonstrates the mathematical necessity of with convincing
rigor.
*
Quaternions (denoted ℍ) have a rich history. Developed in the nineteenth century by Sir William Rowan Hamilton, they fell out of
favor for decades, but have recently enjoyed a renaissance in computer-based models of spatial rotations.

3. Complex Numbers in Quantum Theory
PHY4910 – Spring 2010 – L. White
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In a 1980 paper6
, Wootters says the following:
“By requiring that the distance between states of a general quantum system be determined by the
number of distinguishable intermediate states, could we conclude that the set of states as a whole must
have the geometric structure of the set rays in a complex vector space? [...] if statistical distance is
involved in determining the geometry of this set, it is not the whole story.”
While Wootters' primary focus is not on complex numbers in quantum theory, he nevertheless comments on
the intimate connection between quantum theory and complex space.
More recently, Fuchs7
also references this connection as follows:
“... it is interesting to note that the quantumde Finetti theorem and the conclusions just drawn from it
work only within the framework of complex vector-space quantum mechanics. For quantum mechanics
based on real Hilbert spaces, the connection between exchangeable density operators and unknown
quantum states does nothold.”
Here again, Real Hilbert space cannot accommodate the needs of active development in quantum theory.
Before concluding, it should be noted that neither Saxon nor Stueckelberg addresses Wheeler‟s question “Why
the quantum?” – i.e. they show that there is no quantum theory possible without complex numbers but they do
not attempt to explain why quantum physics has this curious aspect. Wootters makes it clear that a
statistical/information-theoretic approach hints at some underlying necessity but, as he puts it, statistical
distance “is not the whole story.”
Conclusion.
Uncertainty principles did not arrive with quantum mechanics; they were well known and studied in classical
physics. For example, the position and wave number of a classical wave are related by
∆x∆k ≥ 1/2 (6)
It is common to picture this in relation to electromagnetic waves but the same is true in principle for waves in
the ocean; if we know the precise position of the crest of an ocean wave, we cannot know its wave number
with absolute precision, and vice-versa. There is an inbuilt lower limit to the quality of information we can
gather for these two observables at the same instant.
A glance at the Heisenberg uncertainty principle (in this example, for position and momentum) shows an
immediate difference
∆x∆p≥ħ/2 (7)

4. Complex Numbers in Quantum Theory
PHY4910 – Spring 2010 – L. White
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Unity in the numerator is now replaced with the reduced Planck constant h/2π; and the difference is not trivial.
What has changed? In this regard, Saxon is instructive. When we choose to equate the de Broglie wave with
the state function Ψ, we are accepting an uncertainty associated with the wave equation
iħ ∂/∂t[Ψ(x,t)] = - ħ2
/2m ∇2 [Ψ(x,t)] + V(x)Ψ(x,t) (8)
Setting aside the formalism, the core point is this: quantum uncertainty follows unequivocallyfrom wave-particle
duality. If a particle is also a wave then, like all waves, it has an uncertainty. Saxon‟s view might be better
characterized as „wave-particle equivalence‟ rather than wave-particle duality. There is nothing „dual‟ about the
quantum state; it is a wave and it is a particle. Hence Saxon‟s caution about accepting equation (1) above
lightly; there is a lot that comes with it. And, more closely to both Saxon‟s and Stueckelberg‟s arguments, this
particular uncertainty – related as it is to equations (1) and (8) – cannot be derived from the presumption of a
real space. It can only emerge from a mathematical setting which includes i. In moving from the well-
established formalisms of classical physics to their counterparts in the quantum world, we find that complex
numbers cease to be optional; you cannot describe wave or particle behavior at the quantum level without
resort to numbers involving .
And why is this? Because, as stated above, the quantum wave equation describes nothing directly; it serves
only as a predictor of physical phenomena when „squared‟ through multiplication by its complex conjugate. In
classical physics, the wave equation is a direct mathematical description of the physical observable; in
quantum physics, on the other hand, it cannot be. It is rather, as Born first realized, a mathematical description
of something else – the probability amplitude. There is no other useful way to interpret the quantum wave
equation, a conclusionunpalatable to many classical physicists but inescapable nonetheless. So, a wave
equation with a very classical look to it leads us into some strange territory and demands that we let go not
only of classical physics but also our attachment to real numbers. Going back to the ocean wave, it is as if our
mathematics cannot describe the wave itself but only permits us to predict its probable shape from variations in
the quality of light reflected from clouds overhead. Complexity is a natural offshoot of the fact that – at the
quantum level – direct unmediated observation of the wave is no longer a possibility. The thing we observe is
rooted in a space we cannot perceive.

5. Complex Numbers in Quantum Theory
PHY4910 – Spring 2010 – L. White
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Appendix A. Stueckelberg’s Argument in Detail.
Stueckelberg‟s approach to Real Hilbert Space usesreduction ad absurdum – assume a quantum theory built
on ℝ and demonstrate a resulting contradiction. In Boolean terms
(A => B) ^ (B => ¬A) (1)
Here is his reasoning. Observables are symmetric tensors (or symmetric linear operators) in RHS so that
Fab = Fba or FT
= F, GT
= G, ... (2)
The criterion for the impossibility of measuring F and G simultaneously is a non vanishing commutator
FG – GF = [F,G] ≠ 0 (3)
The expectation value <|F,G|> vanishes because [F,G] is an antisymmetric tensor. Only the positive definite
observable
P = -[F,G]2
= PT
(4)
can occur in
<∆F2
><∆G2
>≥ λ2
<P> where λ ∈ ℝ (5)
In order to express the uncertainty principle we introduce the error operators
∆F = F – 1 <F>ψ ; ∆G = G – 1 <G>ψ (6)
from which we form the mean square errors <∆F2
>ψ and <∆G2
>ψ. There are two possibilities
λ2
<P>ψ (7.1)
<∆F2
>ψ<∆G2
>ψ≥
λ2
<C>2
ψ (7.2)
where P is a positive observable of dimension [F]2
[G]2
and C is an observable of dimension [F][G].
Taking equation (7.1) first
<P>ψ = (finite‟)2
> 0 (8)
Now
<∆F2
>ψ = 0 (9)
So
0.(finite)2
≥ λ2
(finite‟)2
(10)
{

6. Complex Numbers in Quantum Theory
PHY4910 – Spring 2010 – L. White
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which has only the trivial solution λ = 0, corresponding to the trivial statement
<∆F2
>ψ<∆G2
>ψ ≥ 0 (11)
Taking equation (7.2) next, we introduce an antisymmetric operator J(FG) and its inverse J-1
(FG), which may be
normalized to -1 and commutes with F and G. This gives rise to an uncertainty principle of the form
<∆F2
>ψ<∆G2
>ψ ≥ λ2
<C(FG)>2
(12)
which requires that J2
(FG)< 0.
So the operator J begins to look like a matrix stand-in for the imaginary unit i. Furthermore, the uncertainty
principle in this second case is of the form
<∆F2
>ψ<∆G2
>ψ ≥ 1/4 <J[F,G]>2
ψ (13)
which is incompatible with Heisenberg.
So, to summarize Stueckelberg: positing a Real Hilbert Space for quantum theory results either in an
uncertainty principle that tells us nothing, or an uncertainty principle that is incommensurate with Heisenberg;
either way, RHS cannot serve as a basis for a quantum theory.
1
Saxon, D.S. “Elementary Quantum Mechanics,” Holden Day (June 1968). ISBN 978-0070549807.
2
Stueckelberg, E.C.G. (1960) “Quantum theory in real Hilbert Space,” Helv. Phys. Acta.33, 727.
3
Singh, J. P., Prabakaran, S., “Quantum Computing Through Quaternions”, EJTP 5 No. 19, 2008.
4
Finkelstein, D., Jauch, J.M., Speiser, D., “Notes on quaternion quantum mechanics.”CERN report 597.
5
Lanczos, C., “On a Field Theoretical Representation of the New Quantum Mechanics”, CLCPPC, Volume III, p. 2-858.
6
Wootters, W. K., “Statistical Distance and Hilbert Space”, Phys. Rev.D 23 No. 2.
7
Fuchs, C. A. “Quantum Mechanics as Quantum Information (and only a little more).” http://arxiv.org/abs/quant-ph/0205039.