Complexification of a Real Krein Space. Part I

Introduction

There is a rather unusual journal called the Journal of Humanistic Mathematics. It aims to provide an open forum for academic and informal discussions of the “human face of mathematics,” focusing on aesthetic, cultural, historical, literary, pedagogical, philosophical, psychological, and sociological aspects of doing, learning, and teaching mathematics for a broad mathematical audience. In Volume 14, Number 2 (July 2024), one finds a very interesting paper, “What is an Imaginary Number? The Plane and Beyond”, by Andrew Powell of Imperial College London [5]. The abstract of that paper reads as follows:

“In this article I argue that $i$ is a quantity associated with the two-dimensional real number plane, whether as a vector, a bi-vector, a point or a transformation (rotation). This position provides a foundation for the complex numbers and accounts for complex numbers in some equations of applied mathematics and physics. I also argue that complex numbers are fundamentally geometrical and can be described by geometric algebra, and that moreover the meaning of complex numbers in physics varies with dimension and geometry of the manifold.
Keywords: complex numbers, geometric algebra, imaginary number, spacetime
algebra, split complex numbers.”

Six of the 43 references in Powell’s article are to papers by David Hestenes, who strongly advocated a geometrical interpretation of the imaginary unit in quantum theory, relating it to a bivector or to a pseudoscalar within geometric (Clifford) algebra [4].

Moretti and Oppio, in their papers [7,8], developed a rather different idea. In Ref.\ [8] they write:

“Relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulations and this complex description is uniquely fixed by physics.”

In [9] Moretti summarizes this point of view as follows:

“An evident issue arises here: why do physicists do not know quantum systems described on real or quaternionic Hilbert spaces? This is a longstanding problem which was recently solved, at least for the physical description of elementary relativistic systems [mor,mor2]. It seems that the complex structure is just a sort of accident imposed by relativistic symmetry.”

At present I am not sure which viewpoint I am willing to adopt (Cf. also [2]). It may well be that several distinct mechanisms are at work simultaneously: Clifford algebra, relativistic symmetry, and signal-theoretic or information-theoretic considerations may all contribute to the privileged role of complex structures in quantum theory. In any case, the practical fact remains that complex structures are remarkably effective and convenient in both mathematics and physics.

Motivated by these considerations, I would like in this series of notes to describe in some detail several complexification procedures, beginning with the elementary complexification of real vector spaces and culminating in the complexification of real Krein spaces. In this first part I collect standard algebraic material in a form suited for later use. The notation is chosen so as to fit naturally with the Krein-space setting that will be discussed in subsequent parts.

For future reference, I shall use the following notation consistently. General real vector spaces will usually be denoted by $X^\mathbb{R}$ , $Y^\mathbb{R}$ , and so on, with their complexifications written as $X^\mathbb{C}$ , $Y^\mathbb{C}$ , etc. When a Krein space enters the discussion, I shall write

$(\mathcal{K}^\mathbb{R},[\cdot,\cdot]_\mathbb{R})$

for a real Krein space and

$(\mathcal{K}^\mathbb{C},[\cdot,\cdot]_\mathbb{C})$

for its complexification. In later parts, a fundamental symmetry on a Krein space will be denoted by $J$ , and the associated Hilbert inner product will be written as

$\langle x,y\rangle_J := [x,Jy].$

This convention makes it possible to pass from the real to the complex setting without changing the basic symbols.

Complexification of a real vector space

I begin by recalling the standard construction of the complexification of a real vector space and of its endomorphisms. Let $X^\mathbb{R}$ be a real vector space. Its complexification $X^\mathbb{C}$ is defined as the tensor product

$X^\mathbb{C} := X^\mathbb{R}\otimes_{\mathbb{R}}\mathbb{C},$

which may be identified with the algebraic direct sum

(1) $\begin{equation*}X^\mathbb{C} = X^\mathbb{R} \oplus iX^\mathbb{R}\end{equation*}$

through the correspondence

(2) $\begin{equation*}x+iy \longleftrightarrow x\otimes 1 + y\otimes i, \qquad x,y\in X^\mathbb{R}.\end{equation*}$

I shall identify $X^\mathbb{R}$ with the real subspace $\{x+i0 : x\in X^\mathbb{R}\}$ of $X^\mathbb{C}$ . Thus one may simply write

$X^\mathbb{C} = X^\mathbb{R} + iX^\mathbb{R}.$

Whenever no confusion is likely, I may also write $X$ instead of $X^\mathbb{R}$ , but the superscripts $\mathbb{R}$ and $\mathbb{C}$ will be kept whenever the distinction matters.

There is a canonical conjugation, or real structure,

$\mathcal{C}: X^\mathbb{C} \to X^\mathbb{C},$

given by

(3) $\begin{equation*}\mathcal{C}(x+iy)=x-iy, \qquad x,y\in X^\mathbb{R}.\end{equation*}$

This map is an antilinear involution, and its fixed-point subspace is precisely $X^\mathbb{R}$ . Thus the pair $(X^\mathbb{C},\mathcal{C})$ may be regarded as a complex vector space equipped with a distinguished real structure from which the original real vector space can be recovered.

It is often convenient to represent vectors in $X^\mathbb{C}$ by columns

$\begin{pmatrix}x\\y\end{pmatrix}, \qquad x,y\in X^\mathbb{R},$

where the column corresponds to the vector $x+iy$ . In this notation, multiplication by $i$ is represented by the matrix

(4) $\begin{equation*}i=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \end{equation*}$

while the conjugation $\mathcal{C}$ is represented by

(5) $\begin{equation*}\mathcal{C}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \end{equation*}$

This elementary block-matrix notation will later reappear in the study of operators and forms on complexified Krein spaces.

Complexification of real linear operators

Let $\mathrm{End}(X^\mathbb{R})$ denote the set of all endomorphisms of $X^\mathbb{R}$ , that is, all real-linear maps from $X^\mathbb{R}$ to itself. This is a real algebra with unit $I_{X^\mathbb{R}}$ . For each $T=T_\mathbb{R}\in\mathrm{End}(X^\mathbb{R})$ I denote by $T_\mathbb{C}$ the unique complex-linear operator

$T_\mathbb{C}: X^\mathbb{C}\to X^\mathbb{C}$

such that

(6) $\begin{equation*}T_\mathbb{C}(ax)=aT_\mathbb{R} x\end{equation*}$

for every $x\in X^\mathbb{R}$ and every $a\in\mathbb{C}$ .

From this defining property one immediately obtains the explicit formula

(7) $\begin{equation*}T_\mathbb{C}(x+iy)=Tx+iTy.\end{equation*}$

In the $2\times 2$ block-matrix notation introduced above, $T_\mathbb{C}$ is represented by

(8) $\begin{equation*}T_\mathbb{C}=\begin{pmatrix}T&0\\0&T\end{pmatrix}.\end{equation*}$

It follows that the map $T\mapsto T^\mathbb{C}$ is a homomorphism of real algebras:

(9) $\begin{equation*}(T_1T_2)^\mathbb{C}=T_1^\mathbb{C} T_2^\mathbb{C}, \qquad I_{X^\mathbb{R}}^\mathbb{C}=I_{X^\mathbb{C}}.\end{equation*}$

A general real-linear operator on $X^\mathbb{C}$ is represented by a matrix of the form

(10) $\begin{equation*}\mathcal{T}=\begin{pmatrix}A&B\\C&D\end{pmatrix},\end{equation*}$

where $A,B,C,D\in\mathrm{End}(X^\mathbb{R})$ . It is useful to decompose such an operator into parts that are even and odd with respect to the conjugation $\mathcal{C}$ . To this end, define

(11) $\begin{align*}\mathrm{Re}(\mathcal{T})&:=\frac12\bigl(\mathcal{T}+\mathcal{C}\mathcal{T}\mathcal{C}^{-1}\bigr),\\ \mathrm{Im}(\mathcal{T})&:=\frac{1}{2i}\bigl(\mathcal{T}-\mathcal{C}\mathcal{T}\mathcal{C}^{-1}\bigr).\end{align*}$

Then

(12) $\begin{equation*}\mathcal{T}=\mathrm{Re}(\mathcal{T})+i \mathrm{Im}(\mathcal{T}).\end{equation*}$

Here $\mathrm{Re}(\mathcal{T})$ is $\mathcal{C}$ -invariant, whereas $\mathrm{Im}(\mathcal{T})$ is $\mathcal{C}$ -anti-invariant.

For $\mathcal{T}$ to be complex-linear, it must commute with $i$ , or equivalently satisfy

$\mathcal{T}i=i\mathcal{T}.$

A straightforward calculation shows that this happens if and only if $D=A$ and $C=-B$ . Thus $\mathcal{T}$ is complex-linear if and only if its real matrix representation is of the form

(13) $\begin{equation*}\mathcal{T}=\begin{pmatrix}A&B\\-B&A\end{pmatrix}.\end{equation*}$

In that case one finds

(14) $\begin{equation*}\mathrm{Re}(\mathcal{T})=\begin{pmatrix}A&0\\0&A\end{pmatrix}, \qquad \mathrm{Im}(\mathcal{T})=\begin{pmatrix}0&B\\-B&0\end{pmatrix}.\end{equation*}$

Thus a complex-linear operator $\mathcal{T}\in\mathrm{End}(X^\mathbb{C})$ is the complexification of a real-linear operator on $X^\mathbb{R}$ if and only if $\mathrm{Im}(\mathcal{T})=0$ .

Remark. In later parts, when $X^\mathbb{R}$ carries additional structure—for instance a symmetric bilinear form or a Krein-space inner product—one will be particularly interested in operators that preserve that structure. The present algebraic analysis of real-linear and complex-linear operators is meant to serve as a template for that more structured setting.

Complexification of real bilinear forms

Ultimately I am interested in indefinite inner products on real vector spaces, and especially on real Krein spaces. For that reason it is useful to recall how a real bilinear form gives rise to a sesquilinear form on the complexification.

Let $\Phi^\mathbb{R}$ be a real bilinear form on $X^\mathbb{R}$ . Adapting a standard construction (see, for example, [bourbaki, Proposition 2, p. 15]) to the present setting, one obtains the following statement.

Proposition. Let $\Phi^\mathbb{R}$ be a real bilinear form on $X^\mathbb{R}$ . Then there exists a unique sesquilinear form $\Phi^\mathbb{C}$ on $X^\mathbb{C}$ such that

(15) $\begin{equation*}\Phi^\mathbb{C}(ax,by)=\overline{a} b \Phi^\mathbb{R}(x,y) \end{equation*}$

for all $a,b\in\mathbb{C}$ and all $x,y\in X^\mathbb{R}$ .

In particular, for $x,y\in X^\mathbb{R}$ one has

(16) $\begin{equation*}\Phi^\mathbb{C}(x,y)=\Phi^\mathbb{R}(x,y).\end{equation*}$

Remark. Throughout these notes I use the physicists’ convention: a sesquilinear form is antilinear in the first argument and linear in the second.

Using Eq. (15), one obtains the explicit formula for $\Phi^\mathbb{C}$ , which may also be taken as its definition:

(17) $\begin{equation*}\Phi^\mathbb{C}(x_1+iy_1,x_2+iy_2)=\Phi^\mathbb{R}(x_1,x_2)+\Phi^\mathbb{R}(y_1,y_2) +i\bigl(\Phi^\mathbb{R}(x_1,y_2)-\Phi^\mathbb{R}(y_1,x_2)\bigr). \end{equation*}$

This formula makes transparent how the original real bilinear form extends to a sesquilinear form on the complexified space.

Using Eq. (17) one obtains the following corollary.

Corollary. The form $\Phi^\mathbb{C}$ is Hermitian, that is,

$\overline{\Phi^\mathbb{C}(z,z')}=\Phi^\mathbb{C}(z',z) \quad\text{for all } z,z'\in X^\mathbb{C},$

if and only if $\Phi^\mathbb{R}$ is symmetric. If $\Phi^\mathbb{R}$ is symmetric, then $\Phi^\mathbb{C}$ is non-degenerate if and only if $\Phi^\mathbb{R}$ is non-degenerate.

Remark. If $\Phi^\mathbb{R}$ is a non-degenerate symmetric bilinear form of indefinite signature on $X^\mathbb{R}$ , then its complexification $\Phi^\mathbb{C}$ is a non-degenerate Hermitian form, generally still indefinite. In the Krein-space setting this is precisely the kind of passage that will later be used to obtain a complex Krein inner product from a real one.

Exercise. Prove the above corollary. As a hint, first verify the Hermitian property directly from Eq. (17), and then analyze non-degeneracy by writing $z=x+iy$ and separating the real and imaginary parts.

References

Banks, B.W., “Time Derivatives of Observables and Applications”, Ph.D. Thesis, 1975. https://ir.library.oregonstate.edu/downloads/ws859h62n
Baez, J., “Division Algebras and Quantum Theory”, Found. Phys. 42, 819–855 (2012). https://arxiv.org/abs/1101.5690
Bourbaki, N., Éléments de Mathématique I. Les structures fondamentales de l’analyse, Livre II: Algèbre, Chapitre 9. Formes sesquilinéaires et formes quadratiques, Actualités Scientifiques et Industrielles 1272, Hermann, Paris, 1959.
Hestenes, D., “Clifford Algebra and the Interpretation of Quantum Mechanics”, in Clifford Algebras and their Applications in Mathematical Physics, edited by J. S. R. Chisholm and A. K. Common, Reidel, 1986. https://davidhestenes.net/geocalc/pdf/caiqm.pdf
Powell, A.W., “What is an Imaginary Number? The Plane and Beyond”, Journal of Humanistic Mathematics 14(2) (2024), 264–285. https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=2051&context=jhm
Li, Bingren, Real Operator Algebras, World Scientific, 2003.
Moretti, V., Oppio, M., “Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry”, Reviews in Mathematical Physics 29, 1750021 (2017). https://arxiv.org/abs/1611.09029
Moretti, V., Oppio, M., “Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one”, Reviews in Mathematical Physics 31, 1950013 (2019). https://arxiv.org/abs/1709.09246
Moretti, V., Fundamental Mathematical Structures of Quantum Theory (Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation), Springer, 2019.

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5 Responses to Complexification of a Real Krein Space. Part I

Bjab says:

2026/06/02 at 10:34

In formula (4)
0 1
-1 0 ->

0 -1
1 0

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- arkajad says:
  
  2026/06/02 at 10:47
  
  Thanks!
  
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Bjab says:

2026/06/02 at 13:54

In formula (11):

Im(T) is always zero ? (Is CTC^-1 always T ?)

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- Bjab says:
  
  2026/06/02 at 14:09
  
  No, no. Zaćmienie.
  
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  - arkajad says:
    
    2026/06/02 at 16:20
    
    The next solar eclipse is scheduled for August 12, 2016. But it will probably be cloudy in my area. If not, I will post a photo.
    
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Introduction

Complexification of a real vector space

Complexification of real linear operators

Complexification of real bilinear forms

Further reading on complexification: Refs. [1,3,6]

References

5 Responses to Complexification of a Real Krein Space. Part I

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