Fundamental Symmetries in Minkowski Krein Space

Posted on 2026/05/20 by arkajad

It is always good to have at least one example. I am reading the Preface to A Course in Functional Analysis by John B. Conway (Springer, 1985), where the author notes:

Historically, mathematics has gone from the particular to the general—not the reverse. There are many reasons for this, but certainly one reason is that the human mind resists abstractions unless it first sees the need to abstract.

The concept of a Krein space is an abstraction, and in this series I am developing this abstraction. Why do we need it? For what purpose? I have several examples in mind. One is the Minkowski space of special relativity, which is a real Krein space. In my previous posts we encountered Clifford algebras and complex structures, in particular complex spaces with Hermitian scalar products of signatures (1,1) and (2,2), which arise naturally when studying conformal symmetries.

In this post we will take a closer look at the Minkowski space introduced in Example 1 and Example 2 in Krein spaces – first steps. I will assume that the reader has done the exercises following that example.

Minkowski space.

Let X=\mathbb{R}^{3,1} be standard Minkowski spacetime with coordinates x^1,x^2,x^3,x^4=ct. Let with G be its metric tensor

    \[ G=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1\end{pmatrix}\]

and the Minkowski scalar product

    \[\langle x,y\rangle =x^TGy=x^1y^1+x^2y^2+x^3y^3-x^4y^4.\]

Then J=G is a fundamental symmetry: we define

    \[(x,y)_J=\langle x,Jy\rangle =x^TGJy=x^Ty,\]

and since J=G, and G^2=I. we obtain

    \[ (x,y)_J=x^1y^1+x^2y^2+x^3y^3+x^4y^4.\]

This is the usual construction turning the indefinite Minkowski form into a positive-definite inner product via a fundamental symmetry. See e.g.  “Krein-Space Framework: Theory & Applications“.

Let L be any Lorentz matrix, that is a real 4\times 4 matrix satisfyng

    \[L^TGL=G.\]

If we define J'=LJL^{-1}, then J'=LL^TG, and J' is again a fundamental symmetry. Thus Lorentz transformations (in particular Lorentz boosts)  generate, in general, new fundamental symmetries.

We now make next step and specify L. First notice that for pure spatial rotation the matrix L is orthogonal, L^TL=I. Therefore for pure rotations we do not obtain a new fundamental symmetry J', because in that case J'=J.  To obtain genuinely new fundamental symmetries we must include Lorentz boosts—transformations to reference frames moving with nonzero velocity with respect to the original one. (cf. Lorentz Transformations in Special Relativity).

Let us think about the physical significance of this observation. What does a fundamental symmetry of a Krein space do? It replaces an indefinite metric by a positive-definite one: in our particular case it replaces Minkowski geometry by a Euclidean geometry. This is equivalent to making the time coordinate imaginary instead of real. But the rate of time depends on the speed of the observer—hence the “twin paradox” and related effects. The splitting of spacetime into “space” and “time” is different for observers in relative motion. Fundamental symmetries precisely encode such splittings; they pick out different space–time decompositions compatible with the same underlying indefinite metric.

In quantum theory, replacing real time by imaginary time—replacing spacetime geometry by Euclidean geometry—is usually called “Wick rotation”. Stephen Hawking used this technique extensively in his work on cosmology. It is sometimes very helpful, but one must be careful, because the resulting Euclidean geometry depends on the observer’s reference frame. Peter Woit has devoted several posts to this subject on his most interesting blog Not Even Wrong:

A pure boost

Let us choose L to be a pure boost in the z– direction:

    \[ L=L(\zeta)=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cosh\,\zeta&\sinh\,\zeta\\ 0&0&\sinh\,\zeta&\cosh\,\zeta\end{pmatrix} \]

Note that \zeta=\tanh^{-1}\beta,\,\beta=v/c. The parameter \zeta is sometimes called the rapidity.

Let us compute J'=LL^TG. In our case L is symmetric, L^T=L, so L^TL=L^2. The matrices L(\zeta) form a one-parameter group,

    \[ L(\zera_1)L(\zeta_2)=L(\zeta_1+\zeta_2),\]

as can be verified by explicit calculations with \sinh and \cosh. Therefore

    \[ L^TL=l(2\zeta)=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cosh\,2\zeta&\sinh\,2\zeta\\ 0&0&\sinh\,2\zeta&\cosh\,2\zeta\end{pmatrix} .\]

This J'=LL^TG is given by

    \[J'=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cosh\,2\zeta&-\sinh\,2\zeta\\ 0&0&\sinh\,2\zeta&-\cosh\,2\zeta\end{pmatrix} .\]

In Krein spaces – a quantum-theoretical monad method, Proposition 2 sates that JJ' and J'J are positive self-adjoint operators on both X_J and X_{J'}. Let us see how this works in our particular case.

The space X_J is \mathbb{R}^4 endowed with the standard Euclidean scalar product, as we have seen it above. What is X_{J'} in our case? The scalar product in X_{J'} is defined by

    \[ (x,y)_{J'}=\langle x,J'y\rangle =x^TGL^TLGy.\]

Using J′=LL^TG and the Lorentz property L^T GL=G, we can rewrite this as

    \[(x,y)_{J'}=x^T GLL^TGy=(LGx)^T LGy.\]

This shows directly that (\cdot,\cdot)_{J′}​ is positive definite: setting z=L^T y. We then have:

    \[ (x,x)_{J'}=z^T z\geq 0,\]

with equality only for z=0, hence only for x=0. This positivity property is not so obvious  if we look  at the explicit formula, which can be computed  as

    \begin{align*} &(x,x)_{J'}=\\&(x^1)^2+(x^2)^2+((x^3)^2+(x^4)^2)\cosh \,2\zeta-2x^3x^4\sinh\,2 \zeta,\end{align*}

containing apparently “dangerous” mixed term that  can be  negative.

The operator T=JJ'

Let us denote

    \[ T= JJ'.\]

Since J^2=J'^2=I, we have

    \[ J'J = T^{-1}.\]

(Indeed,T=JJ' implies T^{-1}=J'^{-1}J^{-1}=J'J..)

Spectral decomposition

We know that T and  T^{-1} are self-adjoint positive operators on both X_J and X_{J'}. Let us analyze their spectral decomposition.

Linear algebra textbooks typically discuss the spectral theorem only in complex spaces, because we want to have n solutions for the eigenvalue equation of an n\times n matrix, and this property of polynomials is guaranteed  over \mathbb{C}, but may fail over \mathbb{R}. A notable exception is  S. Axler, “Linear Algebra Done Right“, where, on p. 136, 2nd edition, we find

Real Spectral Theorem: Suppose that V is a real inner-product space and T\in\mathcal{L}(V). Then V has an orthonormal basis consisting of eigenvectors of T if and only if T is selfadjoint.

Our T is self-adjoint (indeed, positive), so we should be able to find four eigenvalues and four eigenvectors. Moreover, these eigenvectors are orthogonal both in X_J and in X_J'.

Finding the eigenvectors of T is a simple exercise. Here is one convenient choice:

    \[ v_1=\begin{pmatrix}1\\0\\0\\0\end{pmatrix}, v_2=\begin{pmatrix}0\\1\\0\\0\end{pmatrix}, v_3=\begin{pmatrix}0\\0\\1\\1\end{pmatrix}, v_4=\begin{pmatrix}0\\0\\-1\\1\end{pmatrix}.\]

Exercise 1. Verify the above statement.

Their norms in X_J are 1,1,\sqrt{2},\sqrt{2}. To find their norms in X_{J'}, a short computation is needed. The result is:

    \[ \Vert v_1\Vert_{J'}=\Vert v_2\Vert_{J'}=1.\]

    \[ \Vert v_3\Vert_{J'}=\sqrt{2}\,e^{-\zeta},\, \Vert v_4\Vert_{J'}=\sqrt{2}\,e^{\zeta}.\]

Light cone surprise

Eigenvectors v_3 and v_4, corresponding to eigenvalues \neq 1, are on the light cone of Minkowski space: they are isotropic vectors for the Minkowski scalar product \langle \cdot,\cdot\rangle .  When I discovered this fact while writing this post, I was taken by surprise and asked myself: “Why?”. Is it a general feature valid in any Krein space, or is it just a coincidence specific to our example?

It is not difficult to find a simple argument suggesting that it must be so in general. However, because the argument is so simple, I am still suspicious about its validity. Here it is.

Proposition 1 Eigenvectors of JJ' corresponding to eigenvalues \lambda\neq 1 are always isotropic vectors in (X,\langle \cdot,\cdot\rangle ).

Proof. Suppose JJ'v=\lambda v with \lambda\neq 0. Then, since J'J=(JJ')^{-1}  J'Jv=(1/\lambda) v, we have

    \[ \langle v,v\rangle =\langle JJ'v,J'Jv\rangle .\]

Now we use the fact that J and J' are Hermitian with respect to the indefinite scalar product:

    \[\langle Jx,y\rangle =\langle x,Jy\rangle ,\quad \langle J'x,y\rangle =\langle x,J'y\rangle .\]

Thus

    \begin{align*} \langle v,v\rangle &=\langle JJ'v,J'Jv\rangle =\langle J'JJ'v,Jv\rangle =\langle JJ'JJ'v,v\rangle \\&=\lambda^2\langle v,v\rangle .\end{align*}

Since, by assumption, \lambda^2\neq 1, it follows that \langle v,v\rangle =0.

So, at least formally, eigenvectors associated with eigenvalues \lambda\neq 1 are isotropic.

Exercise 2. Is the above “proof” valid for a general Krein space? (I am not sure about the answer).

What next?

Next interesting, finite-dimensional, case of interest is \mathbb{C}^{2,2} – the space of conformal twistors.

However, my aim is to develop a general theory that also covers the case of infinitely many dimensions, both real and complex. In infinite dimensions new complications arise. The spectrum of a bounded self-adjoint operator may be purely continuous, with no eigenvectors of finite norm. Topology (weak, strong, norm) starts to play a role in an essential way. I also want to cover non-separable Hilbert spaces, just in case.

We then need to use full spectral theory, which is usually developed in detail only in the complex case; the real case is often treated more briefly. To bridge this gap we will need to introduce complexification. So there is still some work to be done. In the next post we will discuss the complexification machinery.

After note (20-05-26 13:50)

I think I have found a better, more elegant proof of Proposition 1. The new proof should be adaptable to the continuous spectrum case in infinite dimensions. For a finite dimension it goes as follows: first we prove a simple lemma:
Lemma. JTJ = T^{-1}.
Proof. JTJ=JJJ’J=J’J=T^{-1}.

Then comes corollary:

Corollary. If v is an eigenvector to the eigenvalue \lambda, the Jv is an eigenvector to the eigenvalue 1/\lambda.
Proof. Follows immediately from the lemma.

Now we prove Proposition 1: Assume v is an eigenvector of T to an eigenvalue \lambda\neq 1. We have

    \[ \langle v,v\rangle =(v,Jv)_J=0\]

since vectors v and Jv are eigenvectors belonging to two different eigenvalues, and, by the Spectral Theorem,  are orthogonal in X_J.

Posted in Krein spaces, Linear Algebra, Lorentz transforations, Uncategorized | Tagged , , , , , | 9 Comments

Krein spaces – a quantum-theoretical monad method

Posted on 2026/05/12 by arkajad

In the previous post, Krein spaces – first steps, we introduced the basic definitions and simple examples of Krein spaces, emphasizing how a fundamental symmetry turns an indefinite inner product into a Hilbert space structure. In this post we move from isolated examples to the systematic study of the whole family of Hilbert space structures generated by a given Krein space, and we begin to explore their geometric and physical interpretation.

In the last post we introduced Krein spaces. They are natural generalizations of Hilbert spaces. A single Krein space generates a whole family of Hilbert space scalar products on one and the same vector space. In what follows we will study properties of these Hilbert spaces, examine the geometry of the family, and analyze the transitions between different members of the family. Why is this important? One motivation is explained, for instance, in the monograph Geometrophysics (in Russian) by Yu. S. Vladimirov.

“Geometrophysics / Yu. S. Vladimirov. — 6th ed., electronic. — Moscow: Laboratory of Knowledge, 2024. — 543 pp. — System requirements: Adobe Reader XI; screen 10″. — Title from title screen. — Text: electronic.
ISBN 978-5-93208-696-4

The book is devoted to the exposition and analysis of the geometric approach to the description of the physical world, in particular Einstein’s general theory of relativity and multidimensional geometric theories of physical interactions. The first part provides an introduction to general relativity. The second part examines in detail relativity theory, its formulations, and its generalizations. The third part is devoted to the presentation of a multidimensional geometric theory of the microworld. The fourth part offers a metaphysical analysis of the geometric and other approaches to physics with the aim of substantiating the need to move toward a more advanced picture of the world.”

The book is intended for students and university instructors in physics and mathematics, as well as theoretical physicists and philosophers.

In Chapter 3, “Monad method for describing reference frames,” he examines the concept of a reference frame. He writes:

“Metaphysical analysis (see [41]) shows that any physical theory deals with elements (material entities or events) of three closely interrelated kinds: the objects (events) under consideration, the world surrounding these objects, and the reference body (system).
(…) First of all, let us clarify what an observer (reference body) is and what minimal measuring apparatus it possesses. The analysis shows that such a minimum may be taken to consist of the presence of (proper) clocks and the ability to measure the temporal components of tensor quantities, separating them from the other, spacelike components. We emphasize that the observer has these capabilities precisely at its location, i.e., along its timelike worldline.

As for other points of the manifold, the commonly used device is employed here—one assumes a continuum of observers, each of whom receives and processes information along its own worldline.”

Our “fundamental symmetries” in Krein spaces play exactly this role: they define “reference frames” of the “observer”. As we are dealing with fields “observed” by “extended observers”, each requiring a congruence of world lines,  we need to set our “reference frames” in infinite-dimensional spaces, not just in spacetime of special or general relativity. In a sense, we are extending here the monad method advocated by Vladimirov to a quantum-theoretical framework.

In the previous post Krein spaces – first steps I mentioned that I will assume a basic knowledge of Hilbert spaces. In fact we will immediately need one more advanced result from functional analysis, namely the Hellinger-Toeplitz Therem (see e.g. Michael Mueger, Introduction to Functional Analysis, lecture notes 2024, p. 62) )

THEOREM (Hellinger-Toeplitz (1928))

If H,K are Hilbert spaces and A:H\rightarrow K,\, B:K\rightarrow H are linear maps satisfying

    \[(Ax,z)=(x,Bz)\]

for all x\in H, z\in K, then A and B are bounded.

Note: The scalar product on the left is that of K, that on the right is that of H.

In particular, if H=K and B=A, we obtain the result that any self-adjoint operator defined on the whole  Hilbert space is bounded. This is how this theorem, as a corollary of the Closed Graph Theorem for Banch spaces,  is stated on p. 94 in  “Methods of Modern Mathematical Physicis, vol I: Functional Analysis” by M. Reed and B. Simon, Academic Press 1980.

Let us recall that a linear operator A between two normed spaces is bounded if its norm, defined by

    \[ \Vert A\Vert = \sup_{\Vert x\Vert=1}\Vert Ax\Vert,\]

is finite. An equivalent condition is that there exists a constant C\geq 0 such that

    \[\Vert Ax\Vert\leq C\Vert x\Vert\]

for all x. Then \Vert A\Vert is the smallest such constant. Moreover, A is bounded if and only if it is continuous, if and only if it is continuous at x=0, that is if and only if \Vert x_n\Vert\rightarrow 0 implies \Vert Ax_n\Vert\rightarrow 0. A Hilbert space is a particular case of a Banach space, which is a particular case of a normed space., so the above applies to Hilbert space operators in particular.

Notation: Let now (X,<\cdot,\cdot>) be a Krein space,. We use letters x,y,z,... to denote vectors in X.  Let \mathcal{J}(X) be the space of its fundamental symmetries – see Krein spaces – first steps. We denote by (x,y)_J the Hilbert space scalar product

    \[ (x,y)_J:=<x,Jy>,\]

and by X_J the Hilbert space (X,(\cdot,\cdot)_J). We denote by L(X_J) the algebra of all bounded linear operators on X_J. For any operator A\in L(X_J) we denote by A^J its Hermitian adjoint, defined by

    \[ (Ax,y)_J=(x,A^Jy)_J.\]

Proposition 1. For any J\in\mathcal{J}(X), the operator J is a bounded self-adjoint operator on X_J:

    \[ (x,Jy)_J=(Jx,y)_J.\]

Exercise 1. Use the Hellinger-Toeplitz Theorem to prove the above Proposition 1.

Poposition 2. For any pair J,J'\in \mathcal{J}(X) the operators JJ' and J'J are self-adjoint on both X_J and X_{J'}. Moreover, they are not only self-adjoint, but also positive in X_J and in X_{J'}.

Exercise 2. Prove Proposition 2.

Definition. In a normed space two norms \Vert\cdot\Vert_1 and \Vert\cdot\Vert_2 are said to be equivalent, if there exist constants C_1,C_2>0 such that

    \[\Vert x\Vert_1\leq C_1 \Vert x\Vert_2,\text{ and } \Vert x\Vert_2\leq C_2 \Vert x\Vert_1\]

for all x. When two norms are equivalent, they define the same topology: a sequence x_n converges in one norm if and inly if it converges in the other norm.

Proposition 3. For any J,J'\in\mathcal{J}(X) the Hilbert space norms for X_J and X_{J'} are equivalent.

Exercise 3. Prove Proposition 3.

Corollary 1. For any J,J'\in\mathcal{J}(X), the spaces of bounded linear operators L(X_J) and L(X_{J'}) coincide.

Definition  (of L(X) and A^*)

Using the above corollary we will simply write L(X),  and use the terms “bounded” or “continuous” without specifying the particular Hilbert norm we have in mind. The adjoint of an operator A\in L(X), with respect to the Krein scalar product will be denoted A^*. Thus

    \[ <Ax,y>=<x,A^*y>\]

and

    \[<x,Ay>=<A^*x,y>.\]

Exercise 4. Verify that A^J=JA^*J and A^*=JA^JJ.

Definition (the unitary group f \mathcal{U}(X))

We denote by \mathcal{U}(X) the group of unitary operators in L(X(:

    \[\mathcal{U}(X)=\{U\in L(X): U^*=U^*U=I.\]

Exercise 5. Show that if J is in \mathcal{J}(X) and U is in \mathcal{U}(X), then J':=UJU^* is also in \mathcal{J}(X).

 

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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.

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