Krein spaces – first steps

Posted on 2026/05/09 by arkajad

It was the summer of 1964 (15 June–15 July). The small village of Katsiveli, in the Yalta region of Crimea, hosted the Second Summer School of Mathematics organized by the Ukrainian Academy of Sciences.

Second Summer School of Mathematics, Katsiveli, June-July 1964. In the first row (behind the cat), from left to right, A.N. Kolmogorov, N.N. Bogolyubov. M.G. Krein.

One of the lecture series was delivered by M.G. Krein, under the title Introduction to the Geometry of Indefinite J-Spaces and to the Theory of Operators in These Spaces. Here is the first page of the printed lectures:

Over 77 pages, Krein develops the theory of what he calls “J-spaces.” Today, these are known as Krein spaces, and it is precisely these objects that will concern us here: a particular class of infinite-dimensional spaces endowed with an indefinite metric. They generalize Hilbert spaces—but in a somewhat dangerous direction. Physicists tend to treat them with caution, and often avoid them altogether. Used incautiously in quantum theory, they may lead to “negative probabilities,” which sounds unsettling enough to conjure up “ghosts.” Physicists, as a rule, prefer to avoid ghosts—or to exorcise them as quickly as possible.

But here we are free to play with mathematics: with algebra and geometry in infinite dimensions, where much can be learned. I do, of course, have applications in mind. I even entertain the idea of applying the mathematics of Krein spaces to consciousness. Physicists—Roger Penrose being a notable exception—tend to shy away from consciousness as well, since no widely accepted way of “measuring” it has yet been found. So let us now turn to Krein spaces, as I see them.

Let X be a complex vector space. While we assume here that X is complex, with minor changes everything below also works if X is a real vector space. The space X can have any dimension; it makes no difference for what follows whether the dimension is countable or uncountable. I will assume a basic knowledge of Hilbert spaces, such as is needed in applications to quantum mechanics.

We assume that X is endowed with a nondegenerate hermitian form <x,y>. That means <x,y> is linear in y, antilinear in x, and satisfies

    \[ <x,y>=\overline{<y,x>},\]

and

    \[ <x,y>=0\, \forall y\in X\mbox{ implies }x=0.\]

Note. In one of my previous posts, Conformal structure – until the puppy grows up,
we discussed real n–dimensional vector spaces equipped with a nondegenerate symmetric bilinear form of signature (p,q), p+q=n. An example is Minkowski space, with p=3,q=1 (or p=1,q=3). Here we go further by allowing p, or q, or both, to be infinite. We also allow our space to be over the complex numbers rather than only over the reals. The reason is simple: while vectors in Minkowski space span a 4-dimensional space, vector fields on Minkowski space span an infinite–dimensional vector space. If we complexify the fields, we obtain a complex infinite–dimensional space. And fields (green, or yellow, or lavender, pick your favorite color) are very important!

(Image credit: Handy Marks | public domain)

The universe is made of fields rather than “points”.

Definition. A linear operator J:X\rightarrow X is called a fundamental symmetry if

(i) <Jx,y>=<x,Jy> for all x,y\in X.
(ii) <Jx,Jy>=<x,y> for all x,y\in X.
(iii) (x,y)_J:=<x,Jy> is a Hilbert space scalar product on X.

We denote by \mathcal{J}(X) the set of all fundamental symmetries on X.

We call (X,<\cdot,\cdot>) a Krein space if the set \mathcal{J}(X) of all fundamental symmetries has more than one element.

We denote by X_J the space X endowed with the scalar product (x,y)_J, and by Vert x \Vert_J the corresponding norm:

    \[ \Vert x\Vert_J^2=(x,x)_J.\]

Condition (iii), written explicitly, means that <x,Jx>\,\geq 0 for all x in X, and that it is zero if and only if x=0. Moreover, X_J is complete, in the sense that any sequence (x_n) satisfying

    \[\lim_{m,n\rightarrow\infty} \Vert x_n-x_m\Vert_J=0 \]

— a Cauchy sequence — has a limit x\in X, such that

    \[\lim_{n\rightarrow\infty} \Vert x_n-x\Vert_J=0. \]

Example. Let X=\mathbb{R}^{3,1} be standard Minkowski spacetime, with G its metric tensor

    \[ G=\text{diag}(+1,+1,+1,-1),\]

and the Minkowski scalar product

    \[<x,y>=x^TGy.\]

Then J=G is a fundamental symmetry.

Exercise 1. Verify the above statement.

Exercise 2. Let L be any Lorentz matrix, that is L^TGL+G. Show that if we define J'=LJL^{-1}, then J'=LL^TG, and J' is also a fundamental symmetry. Thus Lorentz boosts generate, in general, new fundamental symmetries.

In the next post we will study basic properties of fundamental symmetries of Krein spaces in more detail.

Print 🖨 PDF 📄 eBook 📱
This entry was posted in Linear Algebra and tagged , . Bookmark the permalink.

8 Responses to Krein spaces – first steps

  1. DMcBride says:
    2026/05/10 at 22:07

    Exercise 2. L^TGL = G

  2. DMcBride says:
    2026/05/11 at 04:08

    Exercise 1.
    (i)Linear operator J:X \mapsto X
    = for all x,y \in X
    Hermitian product in complex vector space X

    in
    X=\mathbb{R}^(3,1)
    with metric tensor G = \begin{bmatrix} +1& 0 & 0 & 0 \\ 0& +1& 0 & 0 \\ 0& 0& +1& 0 \\ 0& 0 & 0 &-1 \\ \end{bmatrix}

    =x^TGy

    x=\begin{bmatrix} x^1\\ x^2\\ x^3\\ x^0 \end{bmatrix} ; x^T=\begin{bmatrix} x^1 &x^2 &x^3 &x^0 \\ \end{bmatrix}
    ;
    y=\begin{bmatrix} y^1 \\ y^2 \\ y^3 \\ y^0 \end{bmatrix};
    x^T(Gy)=x^T\begin{bmatrix} y^1 \\ y^2 \\ y^3 \\ -y^0 \end{bmatrix}=\begin{bmatrix} x^1 &x^2 &x^3 &x^0 \\ \end{bmatrix}
    \begin{bmatrix} y^1 \\ y^2 \\ y^3 \\ -y^0 \end{bmatrix}=x^1y^1+x^2y^2+x^3y^3-x^0y^0
    a scalar value, where
    s=s^T
    so x^TGy=(x^TGy)^T = y^TG^T(x^T)^T; G=G^T, as a symmetric tensor, and (x^T)^T=x; so conclude x^TGy=y^TGx
    The scalar product equivalent to requirement (i) = in the real subspace.

    (ii)
    (iii)

  3. John G says:
    2026/05/11 at 18:17

    Some day I’d like to understand a little better why boson vs ghost and creation vs annihilation seem like the same thing and why position vs momentum and spacetime dimensions vs Kaluza-Klein extra dimensions seem like the same thing. Fock Space vs Krein space seem like the same thing too. While I’m here proper time vs Feynman propagator vs solder form vs affine one form also seem like the same thing.

    Part of the problem is I don’t know enough to appreciate the difference and part of the problem is (semi)classical vs quantum or Feynman vs not Feynman maybe. There was a funny video a while back about a string theorist talking to a Loop Quantum Gravity theorist and they would each talk about a concept in their theory and end up saying together “it’s the same thing” before moving on to another pair of concepts. EEQT has both classical and quantum so this idea can show up in one model when I think a lot about EEQT.

  4. Bjab says:
    2026/05/12 at 14:48

    (iii) (x,y)_J:=
    parenthesis

    Vert x \Vert_J the corresponding ->
    ?

  5. Bjab says:
    2026/05/12 at 14:51

    “Condition (iii), written explicitly, means that \geq 0 for all x in X>”

    How come?

Leave a Reply