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arXiv:quant-ph/9804062v2 21 Oct 1998
LANL xxx E-print archive No. quant-ph/9804062
Fibre bundle formulation of
nonrelativistic quantum mechanics
II. Equations of motion and observables
Bozhidar Z. Iliev∗ † ‡
Short title: Bundle quantum mechanics: II
Basic ideas: → March 1996
Began: → May 19, 1996
Ended: → July 12, 1996
Revised: → December 1996 – January 1997,
Revised: → April 1997, September 1998
Last update: → October 21, 1998
Composing/Extracting part II: → September 27/October 4, 1997
Updating part II: → October 21, 1998
Produced: → February 1, 2008
LANL xxx archive server E-print No.: quant-ph/9804062
BO/•
•HOr ? TM
Subject Classes:
Quantum mechanics; Differential geometry
1991 MSC numbers:
81P05, 81P99, 81Q99, 81S99
1996 PACS numbers:
02.40.Ma, 04.60.-m, 03.65.Ca, 03.65.Bz
Key-Words:
Quantum mechanics; Geometrization of quantum mechanics;
Fibre bundles
∗Department Mathematical Modeling, Institute for Nuclear Research and
Nuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chauss´ ee 72,
1784 Sofia, Bulgaria
†E-mail address: bozho@inrne.bas.bg
‡URL: http://www.inrne.bas.bg/mathmod/bozhome/
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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Contents
I.1Introduction1
I.2Evolution of pure quantum states (review)4
I.3Linear transports along paths and Hilbert fibre bundles6
I.4The Hilbert bundle description of quantum mechanics9
I.5The (bundle) evolution transport 17
I.6Conclusion18
References19
20 This article ends at page . . . . . . . . . . . . . . . . . . . . .
1 Introduction1
2 The bundle equations of motion2
3 The bundle description of observables9
4 Conclusion15
References15
15 This article ends at page . . . . . . . . . . . . . . . . . . . . .
Page 3
Abstract
We propose a new systematic fibre bundle formulation of nonrelativistic
quantum mechanics. The new form of the theory is equivalent to the usual
one but it is in harmony with the modern trends in theoretical physics and
potentially admits new generalizations in different directions. In it a pure
state of some quantum system is described by a state section (along paths)
of a (Hilbert) fibre bundle. It’s evolution is determined through the bundle
(analogue of the) Schr¨ odinger equation. Now the dynamical variables and
the density operator are described via bundle morphisms (along paths). The
mentioned quantities are connected by a number of relations derived in this
work.
In the second part of this investigation we derive several forms of the
bundle (analogue of the) Schr¨ odinger equation governing the time evolution
of state sections. We prove that up to a constant the matrix-bundle Hamilto-
nian, entering in the bundle analogue of the matrix form of the conventional
Schr¨ odinger equation, coincides with the matrix of coefficients of the evolu-
tion transport. This allows to interpret the Hamiltonian as a gauge field.
Here we also apply the bundle approach to the description of observables.
It is shown that to any observable there corresponds a unique Hermitian
bundle morphism (along paths) and vice versa.
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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1. Introduction
This paper is a second part of our investigation devoted to the fibre bun-
dle approach to nonrelativistic quantum mechanics. It is a straightforward
continuation of [?].
The organization of the material is the following.
Sect. 2 is devoted to the bundle analogues of the Schr¨ odinger equa-
tion which are fully equivalent to it. In particular, in it is introduced the
matrix-bundle Hamiltonian which governs the quantum evolution through
the matrix-bundle Schr¨ odinger equation. The corresponding matrix of the
bundle-evolution transport (operator) is found. It is proved that up to a
constant the matrix of the coefficients of the bundle evolution transport co-
incides with the matrix-bundle Hamiltonian. On this basis is derived the (in-
variant) bundle-Schr¨ odinger equation. Geometrically it simply means that
the corresponding state sections are (parallelly, or, more precisely, linearly)
transported by means of the bundle evolution transport (along paths).
In Sect. 3 is considered the question for the bundle description of ob-
servables. It turns out that to any observable there corresponds a unique
Hermitian bundle morphism (along paths) and vice versa.
Sect. 4 closes the work.
The notation of the present work is the the same as the one in [?] and
we are not going to recall it here.
The references to sections, equations, footnotes etc. from [?] are obtained
from their sequential numbers in [?] by adding in front of them the Roman
one (I) and a dot as a separator. For instance, Sect. I.4 and (I.5.13) mean
respectively section 4 and equation (5.13) (equation 13 in Sect. 5) of [?].
Below, for reference purposes, we present a list of some essential equa-
tions of [?] which are used in this paper. Following the just given convention,
we retain their original reference numbers.
i?dψ(t)
dt
i?∂U(t,t0)
∂t
H(t) = i?∂U(t,t0)
∂t
ψ:=?ψ(t)|A(t)ψ(t)?
?ψ(t)|ψ(t)?
ψ(t) = lγ(t)(Ψγ(t)) ∈ F,
?·|·?x= ?lx· |lx·?,
?A‡Φx|Ψx?x:= ?Φx|AΨx?x,
Ψγ(t) = Uγ(t,s)Ψγ(s),
Uγ(t,s) = l−1
γ(t)◦ U(t,s) ◦ lγ(s),
= H(t)ψ(t), (I.2.6)
= H(t) ◦ U(t,t0),
U(t0,t0) = idF, (I.2.8)
◦ U−1(t,t0) = i?∂U(t,t0)
∂t
◦ U(t0,t), (I.2.9)
?A(t)?t
, (I.2.11)
(I.4.1)
x ∈ M, (I.4.4)
Φx,Ψx∈ Fx, (I.4.19)
(I.5.7)
s,t ∈ J. (I.5.10)
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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2.The bundle equations of motion
If we substitute (I.5.11) into (I.2.6)–(I.2.10), we ‘get’ the ‘bundle’ ana-
logues of (I.2.6)–(I.2.10).But they will be wrong!
being that they will contain partial derivatives like ∂lγ(t)/∂t, ∂Ψγ(t)/∂t,
and ∂Uγ(t,t0)/∂t, which are not defined at all. For instance, for the first
one we must have ∂lγ(t)/∂t = limε→0
?1
in this limit is not defined (for ε ?= 0) because lγ(t+ε)and lγ(t)act on dif-
ferent spaces, viz. resp. on Fγ(t+ε)and Fγ(t). The same is the situation
with ∂Uγ(t,t0)/∂t. The most obvious is the contradiction in ∂Ψγ(t)/∂t =
limε→0
?1
ferent (for ε ?= 0) vector spaces.
One way to go through this difficulty is to define, e.g. ∂Ψγ(t)/∂t like
l−1
γ(t)∂ψγ(t)/∂t (cf. (I.4.1)) but this does not lead to something important
and new.
To overcome this problem, we are going to introduce local bases (or
coordinates) and to work with the matrices of the corresponding operators
and vectors in them.
Let {ea(x),a ∈ Λ} be a basis in Fx= π−1(x), x ∈ M. The indices
a,b,c,... ∈ Λ may take discrete, or continuous, or both values. More pre-
cisely, the set Λ has a decomposition Λ = Λd
finite or countable) subsets of N (or, equivalently, of Z) and Λcis union of
subsets of R (or, equivalently, of C). Note that Λdor Λc, but not both, can be
empty. This is why sums like λaea(x) or λaµafor a ∈ Λ, λa,µa∈ C must be
understood as a sum over the discrete (enumerable) part(s) of Λ, if any, plus
the (Stieltjes or Lebesgue) integrals over the continuous part(s) of Λ, if any.
For instance: λaea(x) :=?
By this reason it is better to write
? ?
?
summation convention on indices repeated on different levels.1
The matrices corresponding to vectors or operators in a given field of
bases will be denoted with the same symbol but in boldface, for example:
Uγ(t,s) :=?(Uγ(t,s))a
(Uγ(t,s))a
Analogously, we suppose in F to be fixed a basis {fa(t),
respect to which we shall use the same bold-faced matrix notation, for
instance: U(t,s) =
?Uba(t,s)?, U(t,s)(fa(s)) =: (U(t,s))b
1For details about infinite dimensional matrices see, for instance, [?] and [?, chapter VII,
§ 18]. A comprehensive presentation of the theory of infinite matrices is given in [?]; this
book is mainly devoted to infinite discrete matrices but it contains also some results on
continuous infinite matrices related to Hilbert spaces.
2The matrices U(t,s) and U(t,s) are closely related to propagator functions [?], but
we will not need these explicit connections. For explicit calculations and construction of
mope(t,s) see [?, § 21, §22]
The reason for this
ε(lγ(t+ε)− lγ(t))?, but the ‘difference’
ε(Ψγ(t + ε) − Ψγ(t))?, because Ψγ(t + ε) and Ψγ(t) belong to dif-
?Λcwhere Λdis a union of (a
a∈Λλaea(x) :=?
a∈Λdλaea(x)+?
a∈Λ:=?
a∈Λcλaea(x)da.
a∈Λcda instead of
a∈Λd+?
a∈Λ, but we shall avoid this complicated notation by using the assumed
b
?and Ψγ(s) :=?Ψa
γ(s)?, where Uγ(t,s)(eb(γ(s))) =:
bea(γ(t)) and Ψγ(s) =: Ψa
γ(s)ea(γ(s)).2
a ∈ Λ} with
afb(t), ψ(t) =
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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[ψa(t)], ψ(t) =: ψa(t)fa(t), and, at last, lx(t) =
(lx)b
a(t)fb(t). Generally lx(t) depends on x and t, but if x = γ(s) for some
s ∈ J, we put t = s as from physical reasons is clear that Fγ(t)corresponds
to F at the ‘moment’ t, i.e. the components of lγ(s)are with respect to
{ea(γ(s))} and {fa(s)}. The same remark concerns ‘two-point’ objects like
Uγ(t,s) and U(t,s) whose components will be taken with respect to pairs of
bases like ({ea(γ(t))},{ea(γ(s))}) and ({fa(t)},{fa(s)}) respectively.
Evidently, the equations (I.4.1), (I.5.7)–(I.5.10) remain valid mutatis mu-
tandis in the introduced matrix notation: the kernel letters have to be made
bold-faced, the operator composition (product) must be replaced by matrix
multiplication, and the identity map idFxhas to be replaced by the unit
matrix 1 1Fx:=?δb
and δb
the above definitions, one verifies that (I.5.10) is equivalent to
?(lx)b
a(t)?, lx(ea(x)) =:
a
?:=
?
(idFx)b
a
?
of Fxin {ea(x)}. Here δb
a= 1 for a = b
a= 0 for a ?= b, which means that ea(x) = δb
aeb(x). For instance, using
Uγ(t,s) = l−1
γ(t)(t)U(t,s)lγ(s)(s). (2.1)
Let Ω(x) :=?Ωb
The changes
a(x)?and ω(t) :=?ωb
a(t)?be nondegenerate matrices.
{ea(x)} → {e′
a(x) := Ωb
a(x)eb(x)},
{fa(t)} → {e′
a(t) := ωb
a(t)eb(t)}
of the bases in Fx and F, respectively, lead to the transformation of the
matrices of the components of Φx∈ Fxand φ ∈ F, respectively, according
to
Φx?→ Φ′
x=
?
Ω⊤(x)
?−1Φx,
φ ?→ φ′=
?
ω⊤(t)
?−1φ.
Here the super script ⊤ means matrix transposition, for example Ω⊤(x) :=
??Ω⊤(x)?a
lx(t) ?→ l′
x(t) =
ω⊤(t)
b
?with?Ω⊤(x)?a
b:= Ωa
b(x). One easily verifies the transformation
??−1lx(t)Ω⊤(x) (2.2)
of the components of the linear isomorphisms lx: Fx→ F under the above
changes of bases.
For any operator A(t): F → F we have
A(t) ?→ A′(t) =
?
ω⊤(t)
?−1A(t)ω⊤(t).(2.3)
Analogously, if A(t) is a morphism of (F,π,M), i.e. if A: F → F and
π ◦ A = idM, and Ax:= A(t)|Fx, then
Ax(t) ?→ A′
x(t) =
?
Ω⊤(t)
?−1Ax(t)Ω⊤(t).(2.4)
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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Note that the components of U(t,s), when referred to a pair of bases
{ea(t)} and {ea(s)}, transform according to
U(t,s) ?→ U′(t,s) =
?
ω⊤(t)
?−1U(t,s)ω⊤(s). (2.5)
Analogously, the change {ea(γ(t))} → {e′a(t;γ) := Ωb
nondegenerate matrix Ω(t;γ) :=?Ωb
Uγ(t,s) ?→ U′
γ(t,s) =
a(t;γ)eb(γ(t))}, with a
a(t;γ)?along γ, implies3
Ω⊤(t;γ)
??−1Uγ(t,s)Ω⊤(s;γ). (2.6)
Substituting ψ(t) = ψa(t)fa(t) into (I.2.6), we get the matrix Schr¨ odinger
equation
dψ(t)
dt
= Hm(t)ψ(t) (2.7)
where
Hm(t) := H(t) − i?E(t) (2.8)
is the matrix Hamiltonian (in the Hilbert space description). Here E(t) =
?Eb
dfa(t)/dt = Eb
a(t)fb(t); if fa(t) are independent of t, which is the usual
case, we have E(t) = 0. In the last case Hm= H. It is important to
be noted that Hmis independent of E(t).
the basic vector fa(t), we get H(t)fa(t) = i?[(∂
i?[∂
a(t0,t), that is
a(t)?
determines the expansion of dfa(t)/dt over {fa(t)} ⊂ F, that is
In fact, applying (I.2.9) to
∂tU(t,t0))fb(t0)]Ub
a(t0,t) =
∂t(fc(t)Uc
b(t,t0))]Ub
H(t) = i?∂U(t,t0)
∂t
U(t0,t) + i?E(t)(2.9)
which leads to
Hm(t) = i?∂U(t,t0)
∂t
U(t0,t).(2.10)
Substituting into (2.7) the matrix form of (I.4.1), we find the matrix-
bundle Schr¨ odinger equation
i?dΨγ(t)
dt
= Hm
γ(t)Ψγ(t)(2.11)
where the matrix-bundle Hamiltonian is
Hm
γ(t) := l−1
γ(t)(t)H(t)lγ(t)(t) − i?l−1
γ(t)(t)
?dlγ(t)(t)
dt
+ E(t)lγ(t)(t)
?
. (2.12)
3Cf. [?, equation (2.11)] or [?, equation (4.10)], where the notation L(t,s;γ) =
H(t,s;γ) = Uγ(s,t;γ) and A(t) = Ω⊤(t;γ) is used.
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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Combining (2.8) and (2.12), we find the following connection between
the conventional and bundle matrix Hamiltonians:
Hm
γ(t) = l−1
γ(t)(t)Hm(t)lγ(t)(t) − i?l−1
γ(t)(t)dlγ(t)(t)
dt
. (2.13)
Remark 2.1. Choosing ea(x) = l−1
?δb
H(t) = Hm
γ(t) where we use the dagger (†) to denote also matrix Hermitian
conjugation. Here Hm
γ(t) is a Hermitian matrix in the chosen basis, but in
other bases it may not be such (see (2.23) below). Analogously, choosing
{fa(t)} such that E(t) = 0, we see that Hm(t) = H(t) is a Hermitian
matrix, otherwise it may not be such.
x(fa) for dfa(t)/dt = 0, we get lx(t) =
?Hm
a
?. Then Hγ(t) = H(t). So, as H†= H, we have
γ(t)?†= H†(t) =
Remark 2.2. Note that, due to (2.13), the transition Hm→ Hm
much alike a gauge (or connection) transformation [?] (see also below (2.21)–
(2.23)).
γ is very
Because of (2.11) and (I.5.7) there is 1:1 correspondence between Uγ
and Hm
γexpressed through the initial-value problem (cf. (I.2.8))
i?∂Uγ(t,t0)
∂t
= Hm
γ(t)Uγ(t,t0),
Uγ(t0,t0) = 1 1Fγ(t0), (2.14)
or via the equivalent to it integral equation
Uγ(t,t0) = 1 1Fγ(t0)+1
i?
t ?
t0
Hm
γ(τ)Uγ(τ,t0)dτ.(2.15)
So, if Hm
γis given, we have (cf. (I.2.10))
Uγ(t,t0) = Texp
t ?
t0
1
i?Hm
γ(τ)dτ (2.16)
and, conversely, if Uγis given, then (cf. (I.2.9) and (2.10))4
Hm
γ(t) = i?∂Uγ(t,t0)
∂t
U−1
γ(t,t0) =∂Uγ(t,t0)
∂t
Uγ(t0,t). (2.17)
The next step is to write the above matrix equations into an invariant, i.e.
basis-independent, form. For this purpose we shall use the introduce in [?, ?]
derivation along paths uniquely corresponding to any linear transport along
paths in a vector bundle.
4Expressions like (∂U(t,t0)/∂t)U(t0,t), (∂Uγ(t,t0)/∂t)U−1
are independent of t0 due to [?, propositions 2.1 and 2.4] or [?, propositions 2.1 and 2.4]
(see also (I.3.6) and [?, lemma 3.1]).
γ (t,t0), and U(t,t0)U(t0,t1)
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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Let D be the derivation along paths corresponding to the bundle evo-
lution transport U, that is (cf. [?, definition 2.3] or [?, definition 4.1])
D : γ ?→ Dγ, where Dγ, called derivation along γ generated by U, is such
that Dγ: s ?→ Dγ
s and the derivation
Dγ
s: Sec1?
(F,π,M)|γ(J)
?
→ π−1(γ(s))
along γ at s generated by U is defined by
Dγ
sχ := lim
ε→0
?1
ε[Uγ(s,s + ε)χ(γ(s + ε)) − χ(γ(s))]
?
(2.18)
for any C1section χ over γ(J) in (F,π,M).
By [?, equation (2.27)] or [?, proposition 4.2] the local explicit form
of (2.18) is
Dγ
sχ =
?dχa(γ(s))
ds
+ Γa
b(s;γ)χb(γ(s))
?
ea(γ(s))(2.19)
where the coefficients Γba(s;γ) of U are defined by
Γb
a(s;γ) :=∂ (Uγ(s,t))b
a
∂t
?????t=s
= −∂ (Uγ(t,s))b
a
∂t
?????t=s
. (2.20)
Using (I.5.9) and (2.17), both for t0= t, we see that
Γγ(t) :=
?
Γb
a(t;γ)
?
= −1
i?Hm
γ(t) (2.21)
which expresses a fundamental result: up to a constant the matrix-bundle
Hamiltonian coincides with the matrix of coefficients of the bundle evolu-
tion transport (in a given field of bases). Let us recall that, using another
arguments, analogous result was obtained in [?, sect. 5].
There are two invariant operators corresponding to the Hamiltonian H in
F: the bundle-evolution transport U and the corresponding to it derivation
along paths D. The equations (2.11)–(2.21), as well as the general results
of [?, § 2] and [?, § 4], imply that these three operators, namely H, U, and
D, are equivalent in a sense that if one of them is given, then the remaining
ones are uniquely determined.
Example 2.1. Let {ea(x)} be fixed by ea(x) = l−1
Then Hm
γ(t) is a Hermitian matrix (see remark 2.1). Consequently, in this
case, Γγ(t) is anti-Hermitian, i.e. (Γγ(t))†= −Γγ(t). Note that for other
choices of the bases this property may not hold.
x(fa) for df(t)/dt = 0.
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
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Example 2.2. Let H be given and independent of t, i.e. ∂H(t)/∂t = 0, and
{ea(x)} be fixed by ea(x) = l−1
x(fa) for df(t)/dt = 0. Then lx(t) =
with δb
adefined above. Equations (2.12) and (2.21) yield Hm
and Γγ(t) = −H(t)/i?. Finally, now the solution of (2.14) is Uγ(t,t0) =
exp(H(t)(t − t0)/i?) (cf. (2.16)).
?δb
a
?
γ(t) = H(t)
According to [?, equation (2.30)] (or [?, equation (4.11)]) and foot-
note I.17, if a basis {ea(γ(t))} is change to {e′a(t;γ) = Ωb
with detΩ(t;γ) ?= 0, Ω(t;γ) :=?Ωb
Γ′
a(t;γ)eb(γ(t))}
a(t;γ)?, then Γγ(t) transforms into5
γ(t) = (Ω⊤(t;γ))−1Γγ(t)Ω⊤(t;γ) + (Ω⊤(t;γ))−1dΩ⊤(t;γ)
dt
. (2.22)
This result is also a corollary of (2.5) and (2.20).
Hence (see (2.21)), the matrix-bundle Hamiltonian undergoes the change
Hm
γ(t) where
γ(t) ?→′Hm
′Hm
γ(t) = (Ω⊤(t;γ))−1Hm
γ(t)Ω⊤(t;γ) − i?(Ω⊤(t;γ))−1dΩ⊤(t;γ)
dt
,(2.23)
which can be deduced from (2.13) too.
Now we are able to write into an invariant form the matrix-bundle
Schr¨ odinger equation (2.11). Substituting (2.21) into (2.11) and using (2.19),
we find that (2.11) is equivalent to
Dγ
tΨγ= 0.(2.24)
This is the (invariant) bundle Schr¨ odinger equation (for the state sections).
Since it coincides with the linear transport equation along γ [?, definition 5.2]
for the bundle evolution transport, it has a very simple and fundamental
geometrical meaning. By [?, proposition 5.4] this is equivalent to the state-
ment that Ψγis a (linearly) transported along γ section with respect to the
bundle evolution transport (expressed in other terms via (I.5.7); see [?, defi-
nition 2.2]). Note that (2.24) and (I.5.7) are compatible as [?, equation (4.5)]
is fulfilled (see also [?, equation (2.25)]): Dγ
not a summation index here!). Moreover, if D is given (independently of
U, e.g. through (2.19)), then from [?, proposition 5.4] follows that U is the
unique solution of the (invariant) initial-value problem
t◦ Uγ(t,t0) ≡ 0, t,t0∈ J (γ is
Dγ
t◦ Uγ(t,t0) = 0Uγ(t0,t0) = idFγ(t0). (2.25)
This is the bundle Schr¨ odinger equation for the evolution transport (opera-
tor) U. In fact it is the inversion of (2.18) with respect to U.
Let us summarize in conclusion. There are two equivalent ways of de-
scribing the unitary evolution of a quantum system: (i) through the evolu-
tion transport U (see (I.2.1)) or by the Hermitian Hamiltonian H (see (I.2.6))
5In [?, ?] the matrix A(t) = Ω⊤(t;γ) is used instead of Ω(t;γ).
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
8
in the Hilbert space F (which is the typical fibre in the bundle description)
and (ii) through the bundle evolution transport U (see (I.5.7)), which is
a Hermitian (and unitary) transport along paths, or the derivation along
paths D (see (2.24)) in the Hilbert fibre bundle (F,π,M). In the bun-
dle description U corresponds to U (see (I.5.10)) and D to H (see (2.19)
and (2.21)).
Since now we have in our disposal the machinery required for analysis
of [?], we, as promised in Sect. I.1, want to make some comments on it.
In [?, p. 1455, left column, paragraph 4] is stated “that in the Heisenberg
gauge (picture) the Hamiltonian is the null operator”. If so, all eigenvalues
of the Hamiltonian vanish and, as they are picture-independent, they are
null in any picture of quantum mechanics. Consequently, form here one
deduces the absurd conclusion that the ‘energy levels of any system coin-
cide and correspond to one and the same energy equal to zero’. Since the
paper [?] is mathematically completely correct and rigorous, there is some-
thing wrong with the physical interpretation of the mathematical scheme
developed in it. Without going into details, we describe below the solution
of this puzzle which simultaneously throws a bridge between [?] and the
present investigation.
In [?] the system’s Hilbert space H is replace by a differentiable Hilbert
bundle E(R+,H) (in our terms (E,π,R+) with a fibre H)), R+:= {t : t ∈
R,t ≥ 0}, which is an associated Hilbert bundle of the principle fibre bundle
P?R+,U(H)?
of (linear) bounded invertible operators in H with bounded inverse. Let
p : U(H) → GL(C,dimH) be a (linear and continuous) representation of
U(H) into the general linear group of dimH-dimensional matrices. An obvi-
ous observation is that [?, equation (4.6)] under p transforms, up to nota-
tion, to our equation (2.23) (in [?] is taken ? = 1). Thus we see that what
in [?] is called Hamiltonian is actually the (analogue of the) matrix-bundle
Hamiltonian Hm
γ(t), not the Hamiltonian H itself (see also Sect. 3). This
immediately removes the above-pointed conflict: as we shall see later in the
third part of this series, along any γ (or, over R+in the notation of [?] -
see below), we can choose a field of frames (bases) in which Hm
cally vanishes but, due to (2.12), this does not imply the vanishment of the
Hamiltonian at all. This particular choice of the frames along γ corresponds
to the ‘Heisenberg gauge’ in [?], normally known as Heisenberg picture.
Having in mind the above, we can describe [?] as follows.
have F = E, M = R+, F = H (the conventional system’s Hilbert space),
J = R+, γ = idR+(other choices of γ correspond to reparametrization of
the time), and
the matrix-bundle Hamiltonian Hm
γ(t) represents the operator A(t) of [?],
incorrectly identified there with the ‘Hamiltonian’ and the choice of a field
of bases over γ(J) = R+= M corresponds to an appropriate ‘choice of the
gauge’ in [?]. Now, after its correspondence between [?] and the present
of orthonormal bases of H where U(H) is the unitary group
γ(t) identi-
In it we
∂
∂t, t ∈ R+is the analog of D+in [?]. As we already pointed,
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
9
work is set, one can see that under the representation p the main results
of [?], expressed by [?, equations (4.5), (4.6) and (4.8], correspond to our
equations (2.24) (see also (2.19)), (2.23) and (2.4) respectively.
Ending with the comment on [?], we note two things. First, this paper
uses a rigorous mathematical base, analogous to the one in [?], which is not
a goal of our work. And, second, the ideas of [?] are a very good motivation
for the present investigation and are helpful for its better understanding.
3. The bundle description of observables
In quantum mechanics is accepted that to any dynamical variable, say A A A,
there corresponds a unique observable, say A(t), which is a Hermitian linear
operator in the Hilbert space F, i.e. A(t): F → F, A(t) is linear, and
A†= A [?, ?, ?]. What is the analogue of A(t) in the developed here bundle
description? Below we prove that this is a suitable bundle morphism A of
the introduced in Sect. I.5 fibre bundle (F,π,M).
Let ψ(λ)(t) ∈ F be an eigenvector of A(t) with eigenvalue λ (∈ R),
i.e. A(t)ψ(λ)(t) = λψ(λ)(t).According to (I.4.1) to ψ(λ)(t) corresponds
the vector Ψ(λ)
the Hilbert space and Hilbert bundle descriptions of a quantum evolution
are fully equivalent (see Sect. I.5).
spond certain operator which we denote by Aγ(t). We define this opera-
tor by demanding any Ψ(λ)
γ (t) to be its eigenvector with eigenvalue λ, i.e.
(Aγ(t))Ψ(λ)
γ (t). Combining this equality with the preceding two,
we easily verify that?Aγ(t) ◦ l−1
linearity of lxhas been used. Admitting that {ψ(λ)(t)} form a complete set
of vectors, i.e a basis of F, we find
γ (t) = l−1
γ(t)ψ(λ)(t) ∈ Fγ(t)in the bundle description. But
Hence to A(t) in Fγ(t)must corre-
γ (t) := λΨ(λ)
γ(t)
?ψ(λ)(t) =?l−1
γ(t)◦ A(t)?ψ(λ)(t) where the
Aγ(t) = l−1
γ(t)◦ A(t) ◦ lγ(t): Fγ(t)→ Fγ(t).(3.1)
More ‘physically’ the same result is derivable from (I.2.11) too. The
mean value ?A?t
ψof A at a state ψ(t) is given by (I.2.11) and the mean
value of Aγ(t) at a state Ψγ(t) is
?Aγ(t)?t
Ψγ=?Ψγ(t)|Aγ(t)Ψγ(t)?γ(t)
?Ψγ(t)|Ψγ(t)?γ(t)
, (3.2)
that is it is given via (I.2.11) in which the scalar product ?·|·?x, defined
by (I.4.4), is used instead of ?·|·?. We define Aγ(t) by demanding
?A(t)?t
ψ= ?Aγ(t)?t
Ψγ.(3.3)
Physically this condition is very natural as it means that the observed values
of the dynamical variables are independent of the way we calculate them.
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
10
From this equality, (I.4.1), and (I.4.4), we get ?ψ(t)|A(t)ψ(t)? = ?ψ(t)|lγ(t)◦
Aγ(t)◦l−1
γ(t)ψ(t)? which, again, leads to (3.1). Thus we have also proved the
equivalence of (3.1) and (3.3).
According to equation (3.1), along γ: J → M to any operator A(t): F →
F, t ∈ J, there corresponds a unique map Aγ(t): Fγ(t)→ Fγ(t)in any fibre
Fγ(t), t ∈ J, in (F,π,M). If J′⊆ J is a subinterval on which γ is without self-
intersections and Aγ|J′: π−1(γ(J′)) → π−1(γ(J′)) is defined by Aγ|J′??
Aγ(t′): Fγ(t′)→ Fγ(t′), t′∈ J′, then Aγ|J′ ∈?Morf(F,π,M)|γ(J′)
is a morphism on the restricted on γ(J′) fibre bundle (F,π,M). In the
general case we define the multiple-valued map Aγ: F → F via Aγ|Fx:=
{Aγ(t) : t ∈ J,γ(t) = x} for every x ∈ M. Evidently Aγ|Fx= ∅ for
x ?∈ γ(J) and Aγ(t)|Fγ(t): Fγ(t)→ Fγ(t), t ∈ J, the multiplicity of Aγ|Fγ(t)
being equal to one plus the number of self-intersections of γ at the point
γ(t). We call a (bundle) morphism along paths6any map A: γ ?→ Aγ, where
Aγ: F → F can be multiple-valued and such that π◦
and Aγ|Fx= ∅ for x ?∈ γ(J). We call the (possibly multiple-valued) map Aγ
a (bundle) morphism along the path γ. Hence, the above-defined map Aγis
a morphism along γ which is singled-valued (and consequently a morphism
over γ(J)) iff γ is without self-intersections. Therefore the map A: γ ?→ Aγ
is morphism along paths. We call A Hermitian and denote this by A‡= A,
if Aγare such, i.e. if (I.4.19) holds for Aγinstead of A. The morphism along
paths A is Hermitian if A(t) is a Hermitian operator, viz. we have
Fγ(t′)=
?, i.e. Aγ|J′
?
Aγ|π−1(γ(J))
?
= idγ(J)
A‡= A ⇐⇒ A‡
γ(t) = Aγ(t) ⇐⇒ A†(t) = A(t), (3.4)
which is a simple corollary of (3.1) and (I.4.20). Hence, if the morphism
Aγ(t) along γ corresponds to an observable A, it is Hermitian because A(t)
is such by assumption [?, ?]
Consequently, to any observable A there corresponds a unique Hermitian
bundle morphism A along paths and vice versa. Explicitly this correspon-
dence is given by (3.1) which will be assumed hereafter. Its consequence
is the independence of the physically measurable quantities (and the eigen-
values of the corresponding operators) of the mathematical way we describe
them, as it should be.
Generally to any operator A: F → F there corresponds a unique (global)
morphism A ∈ Morf(F,π,M) given by
Ax= A??Fx= l−1
x ◦ A ◦ lx,x ∈ M,
A: F → F.(3.5)
Consequently to an observable A(t) can be assigned the (global) morphism
A(t), A(t)|Fx= l−1
x ◦A(t)◦lx. But this morphism A(t) is almost useless for
6Cf. the definition of a (bundle) morphism C ∈ Morf(F,π,M) := {B :
F, π ◦ B = idM} of (F,π,M).
B : F →
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
11
our goals as it simply gives in any fibre Fxa linearly isomorphic image of
the initial observable A(t) (see Sect. I.4).
Notice that Aγ(t) generally depends explicitly on t even if A does not.
In fact, from (3.1) we get
∂Aγ(t)
∂t
=?gγ(t),Aγ(t)?
+ l−1
γ(t)(t)∂A(t)
∂t
lγ(t)(t), (3.6)
where [·,·]−denotes the commutator of corresponding quantities and
gγ(t) := −l−1
γ(t)(t)dlγ(t)(t)
dt
. (3.7)
In particular, to the Hamiltonian H in F there corresponds the bundle
Hamiltonian (or the bundle-Hamiltonian morphism along paths) given by
Hγ(t) := l−1
γ(t)◦ H(t) ◦ lγ(t). (3.8)
It is a Hermitian bundle morphism along paths, H‡
tian operator.
From (3.8), using (I.2.9) and (I.5.10), we find
γ = Hγ, as H is a Hermi-
Hγ(t) = i?l−1
γ(t)◦∂U(t,t0)
∂t
◦ lγ(t0)◦ Uγ(t0,t). (3.9)
From here we can get a relationship between the matrix-bundle Hamiltonian
and the bundle Hamiltonian. For this purpose we write (3.9) in a matrix
form and using (2.17) and dfa(t)/dt = Eb
afb(t), we obtain:
Hγ(t) = Hm
γ(t) + i?l−1
γ(t)(t)
?dlγ(t)(t)
dt
+ E(t)lγ(t)(t)
?
. (3.10)
Substituting here (2.12), we get
Hγ(t) = l−1
γ(t)(t)H(t)lγ(t)(t) (3.11)
which is simply the matrix form of (3.8). Combining (3.10) with (2.13), we
find the following connection between the matrix of the bundle Hamiltonian
and the matrix Hamiltonian:
Hγ(t) = l−1
γ(t)(t)Hm(t)lγ(t)(t) + i?l−1
γ(t)(t)E(t)lγ(t)(t). (3.12)
We notice that, due to (3.5) as well as to (3.1), to the identity map of F
there corresponds a morphism along paths equal to the identity map of F:
idF←→ idF. (3.13)
The results expressed by (3.1) and (3.5) enable functions of observables
in F to be transferred into ones of morphisms along paths or morphisms
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
12
of (F,π,M), respectively. We will illustrate this in the case of, e.g., two
variables. Let G: (A(t),B(t)) ?→ G(A(t),B(t)): F → F be a function of the
observables A(t),B(t): F → F. It is natural to define the bundle analogue
G of G by
G : (A,B) ?→ G(A,B): γ ?→ Gγ(A,B): π−1(γ(J)) → π−1(γ(J)),
where Gγ(A,B)|Fx= ∅ for x ?∈ γ(J) and
Gγ(A,B)|Fγ(t):= l−1
γ(t)◦ G(A(t),B(t)) ◦ lγ(t)
= l−1
γ(t)◦ G(lγ(t)◦ Aγ(t) ◦ l−1
γ(t),lγ(t)◦ Bγ(t) ◦ l−1
γ(t)) ◦ lγ(t). (3.14)
Thus G(A,B) is a bundle morphism along paths. This definition becomes
evident in the cases when G is a polynom or if it is expressible as a con-
vergent power series; in both cases the multiplication has to be understood
as an operator composition. If we are dealing with one of these cases, the
definition (3.14) follows from the fact that for any morphisms A1,... ,Ak,
k ∈ N along paths of (F,π,M) the equality
A1,γ(t) ◦ A2,γ(t) ◦ ··· ◦ Ak,γ(t) = l−1
γ(t)◦ (A1(t) ◦ A2(t) ◦ ··· ◦ Ak(t)) ◦ lγ(t)
(3.15)
holds due to (3.1). In these cases G(A,B) depends only on A and B and it
is explicitly independent on the isomorphisms lx, x ∈ M.
The commutator of two operators is a an important operator function
in quantum mechanics. In the Hilbert space and bundle descriptions it is
defined by [A,B] := A◦B−B◦A and [A,B] := A◦B −B ◦A respectively.
The relation
[Aγ(t),Bγ(t)] = l−1
γ(t)◦ [A,B] ◦ lγ(t)
(3.16)
is an almost evident corollary of (3.1). It can also be considered as a spe-
cial case of (3.14). In particular, to commuting observables (in F) there
correspond commuting bundle morphisms (of (F,π,M)):
[A,B] = 0 ⇐⇒ [A,B] = 0. (3.17)
A little more general is the result, following from (3.16), that to observ-
ables whose commutator is a c-number there correspond bundle morphisms
with the same c-number as a commutator:
[A,B] = c(idF) ⇐⇒ [A,B] = c(idF). (3.18)
for some c ∈ C. In particular, the bundle analogue of the famous relation
[Q,P] = i?(idF) between a coordinate Q and the conjugated to it momen-
tum P is [Q,P] = i?(idF).
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
13
A bit more complicated is the case for operators and morphisms along
paths at different ‘moments’. Let γ: J → M and r,s,t ∈ J. If˘Gs,t: (A,B) ?→
G(A(s),B(t)), we define the bundle analogue˘Gs,tof˘Gs,tby
˘Gs,t: (A,B) ?→˘Gs,t(A,B): γ ?→˘Gγ;s,t(A,B): π−1(γ(J)) → π−1(γ(J)),
where
˘Gγ;s,t(A,B)
???Fγ(t)
:= l−1
γ(r)◦ G(A(s),B(t)) ◦ lγ(r)
= l−1
γ(r)◦ G(lγ(r)◦˘ Aγ;s(r) ◦l−1
γ(r),lγ(r)◦˘Bγ;t(r)◦l−1
γ(r)) ◦lγ(r): Fγ(r)→ Fγ(r).
(3.19)
Here
˘ Aγ;t(r) := l−1
γ(r)◦ A(t) ◦ lγ(r)= lγ
t→r◦ A(t) ◦ lγ
r→t: Fγ(r)→ Fγ(r), (3.20)
where (3.1) has been used and lγ
(along paths) from γ(s) to γ(t) assigned to the isomorphisms lx, x ∈ M (see
Sect. I.4, equation (I.4.16)).7Now the analogue of (3.15) is
s→t:= lγ(s)→γ(t)is the (flat) linear transport
˘ A1;γ;t1(r) ◦˘ A2;γ;t2(r) ◦ ··· ◦˘ Ak;γ;tk(r)
= l−1
γ(r)◦ (A1(t1) ◦ A2(t2) ◦ ··· ◦ Ak(tk)◦) ◦ lγ(r). (3.21)
So, if G is a polynom or a convergent power series, the morphism˘Gγ;s,t(A,B)
along γ depends only on˘Aγ;s(r) and˘Bγ;t(r).
In particular for G(·,·) = [·,·] , have
?˘Aγ;s(r),˘Bγ;t(r)
?
= l−1
γ(r)◦ [A(s),B(t)] ◦ lγ(r)
(3.22)
which for s = r = t reduces to (3.16). In this case the analogues of (3.17)
and (3.18) are
[A(s),B(t)] = 0 ⇐⇒
?˘Aγ;s(r),˘Bγ;t(r)
?˘Aγ;s(r),˘Bγ;t(r)
?
?
= 0,(3.23)
[A(s),B(t)] = c(idF) ⇐⇒
= c(idFγ(r)),(3.24)
respectively.
The above considerations can mutatis mutandis, e.g by replacing γ(t)
with x, A(t) with A, A with A, etc., be transferred to global morphisms of
(F,π,M), but this is not needed for the present investigation.
7According to [?, sections 2 and 3] the morphism˘Aγ;t(r) along γ is obtained via linear
transportation of Aγ(t) along γ by means of the induced by lγ
paths in the fibre bundle morf (F,π,M) of bundle morphisms of (F,π,M)
s→tlinear transport along
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
14
Further we will need a kind of ‘extension’ of the differentiation along
paths Dγ
t(see (2.18)) on some fibre morphisms that will be presented below.
Let Morfp(F,π,M), p ∈ N ∪ 0, be the set of Cpbundle morphisms of
(F,π,M). We define˜Dγ
tto be the differentiation along γ (at t) of bundle
morphisms from Morf1(F,π,M) linearly acting on state vectors, mapping
them on Morf0(F,π,M)??γ(J)and given by
˜Dγ
t: C ?→˜Dγ
t(C) := Dγ
t◦ C|γ(t)
(3.25)
where C ∈ Morf1(F,π,M) acts on state vectors (only).
Applying (2.19), we can find the explicit (matrix) action of˜Dγ
Ct:= C|γ(t)and [ [ [X] ] ] be the matrix of a vector or an operator X in {ea}.
Due to (2.19), we have [ [ [˜Dγ
t(C)Ψγ(t)] ] ] =
Γγ(t)CtΨγ(t). Substituting here
dtΨγ(t) from (2.11) and using (2.21), we
obtain the matrix equation
t. Let
?d
dtCt
?Ψγ(t) + Ct
?d
dtΨγ(t)?+
d
[ [ [˜Dγ
t(C)Ψγ(t)] ] ] =
?d
dtCt
?
Ψγ(t) + [Γγ(t),Ct] Ψγ(t), (3.26)
where [·,·] is the commutator of the corresponding matrices, or
[ [ [˜Dγ
t(C)] ] ] =d
dtCt+ [Γγ(t),Ct] .(3.27)
Nevertheless that the last equation is valid in any local basis it cannot
be written in an invariant (operator) form as the action of
morphisms or sections is not defined, as well as to Γγ(t) alone there does
not correspond some invariant operator or morphism.
We derived (3.27) under the assumption that˜Dγ
i.e on ones satisfying the matrix-bundle Schr¨ odinger equation (2.11). Con-
versely, if we apply (3.27) to some vector Φγ(t) ∈ Fγ(t)and compare the
result with the one for (Dγ
t(C))(Φγ(t)) obtained through (2.19) (see above),
we see that Φγ(t) satisfies (2.11). Consequently, equation (3.27) is valid if
and only if it is applied on vectors representing the evolution of a quantum
system. Hence Ψγ(t) is a state vector, i.e. it satisfies, for instance, the bun-
dle Schr¨ odinger equation (2.24), iff in any basis the equation (3.26) is valid
for any bundle morphism C. In particular (3.26) is valid for the (Hermitian)
morphisms (along paths) corresponding to observables and Ψγ(t) satisfying
the bundle Schr¨ odinger equation (2.24).
The over-all above discussion shows the equivalence of (3.26) (for ev-
ery morphisms C) with the Schr¨ odinger equation (in anyone of its (equiva-
lent) forms mentioned until now). That is why (3.26) can be called matrix-
morphism Schr¨ odinger equation.
d
dton bundle
tacts on state vectors,
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Bozhidar Z. Iliev: Bundle quantum mechanics. II
15
4. Conclusion
Here we have continued to apply the fibre bundle formalism to nonrelativistic
quantum mechanics. We derived different forms of the bundle Schr¨ odinger
equation which governs the time evolution of state sections along paths in
the Hilbert bundle description.
In this description, as we have seen, the observables are described via
Hermitian bundle morphisms along paths. We also have concerned some
technical problems connected with functions of observables.
In the future continuation of the present series we plan to consider from
a fibre bundle point of view the following items: pictures and integrals of
motion, mixed states, evolution transport’s curvature, interpretation of the
theory and its possible further developments.
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