Guido Bacciagaluppi
email: guido.bacciagaluppi@balliol.ox.ac.uk

Gauge-and Galilei-Invariant Geometric Phases

Neither geometric phases nor di erences in geometric phases are generally invariant under time-dependent unitary transformations (unlike di erences in total phases), in particular under local gauge transformations and Galilei transformations. (This was pointed out originally by Aharonov and Anandan, and in the case of Galilei transformations has recently been shown explicitly by Sj oqvist, Brown and Carlsen.) In this paper, I introduce a phase, related to the standard geometric phase, for which phase di erences are both gauge-and Galileiinvariant, and, indeed, invariant under transformations to linearly accelerated coordinate systems. I discuss in what sense this phase can also be viewed as geometric, what its relation is to earlier proposals for making geometric phases invariant under gauge or Galilei transformations, and what is its classical analogue. I nally apply this invariant phase to Berry's derivation of the Aharonov{Bohm e ect.

Introduction

The eld of geometric phases has developed out of [START_REF] Berry | Quantum Phase Factors Accompanying Adiabatic Changes[END_REF] seminal paper, in which he identi ed a geometric component of the phase in adiabatic cyclic evolutions. This concept was further developed and generalised in subsequent important papers by B. [START_REF] Simon | Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase[END_REF], [START_REF] Aharonov | Phase Change during a Cyclic Quantum Evolution[END_REF], [START_REF] Samuel | General Setting for Berry's Phase[END_REF], [START_REF] Aitchison | On the Real and Complex Geometric Phase[END_REF], [START_REF] Mukunda | Quantum Kinematic Approach to the Geometric Phase. I. General Formalism[END_REF], and others. I shall be considering in particular the geometric phase as treated by Aharonov and Anandan (AA) for the case of arbitrary cyclic evolutions and its generalisation by Aitchison and Wanelik (AW) to the non-cyclic case.

As was pointed out by [START_REF] Aharonov | Phase Change during a Cyclic Quantum Evolution[END_REF], the geometric phase is neither locally gauge-invariant, nor (as has been shown explicitly by Sj oqvist, [START_REF] Sj Oqvist | Galileian Noninvariance of Geometric Phase[END_REF]) is it Galilei-invariant. In this paper, I shall show how to construct a phase that is closely related to the geometric phase of AA or AW, and such that phase di erences (as also done by Garc a de Polavieja (1997)) become both locally gauge-and Galilei-invariant, as well as invariant under transformations to linearly accelerated frames.

The paper is structured as follows. After recalling the basic de nitions and properties of geometric phases in Section 2, I shall sketch the proofs of non-invariance under local gauge and Galilei transformations (Section 3). In Section 4 I review previous treatments of this non-invariance, in particular by [START_REF] Anandan | Geometric Quantum Phase and Angles[END_REF] and by Garc a de [START_REF] Garc A De Polavieja | Galileian Invariant Structure of Geometric Phase[END_REF]. I introduce an extended geometric phase in Section 5 such that, as shown in Section 6, phase di erences are indeed invariant under local gauge and Galilei transformations and under transformations to linearly accelerated coordinate systems. Section 7 discusses the geometric interpretation of this phase, and Section 8 its classical analogue. Finally, Section 9 applies the extended geometric phase to [START_REF] Berry | Quantum Phase Factors Accompanying Adiabatic Changes[END_REF] calculation of the Aharonov{Bohm (AB) e ect, making his identi cation of the AB phase with a geometric-type phase invariant under the above transformations.

2 Geometric Phases Aitchison and Wanelik (1992, Appendix; see also [START_REF] Mukunda | Quantum Kinematic Approach to the Geometric Phase. I. General Formalism[END_REF]) de ne the geometric phase, which we shall denote by ], of a generally non-cyclic (i. e. open) path : s 2 0; S] 7 ! (j (s)i) in projective Hilbert space (ray space) P(H) as follows (here (j (s)i) denotes the projection of the vector j (s)i onto the ray containing j (s)i, and s is a curve parameter, that is, the mapping is locally one-to-one):1 e i ] := h (0)j (S)i h (S)j (0)i (1)

If one considers only normalised vectors, j j (s)j j = 1, de nition (1) reduces to e i ] := h (0)j (S)i h (S)j (0)i (2) which for the case of a cyclic path ( (0) = (S)) is equivalent to AA's de nition. In this last case, if s 7 ! j (s)i is a single-valued representation in the Hilbert space of the cyclic path , then ] can be further written as ] = i I h (s)j d ds j (s)ids = i Z S 0 h (s)j d ds j (s)ids

(3) (whenever we use the notation H , we shall assume that the representation chosen for the path is single-valued).

The geometric phase (2) can be expressed as the di erence between the total phase ', given by e i' := h (0)j (S)i h (S)j (0)i 1=2 ;

(

and what we shall call the locally accumulated phase , de ned by := i Z S 0 h (s)j d ds j (s)ids:

(5)

The phase (2) has two properties that make it a function of the unparametrised path in ray space. First of all, it is a projective quantity, in the sense that it is independent of the particular (di erentiable) curve s 7 ! j (s)i in Hilbert space that projects to . Another such curve can be written as j ~ (s)i = e if(s) j (s)i ( 6) for some real function f (s). In fact we have (in obvious notation):

e i ;f] = h (0)je if(0) e if(S) j (S)i h (S)je if(S) e if(0) j (0)i ! 1=2 exp ( Z S 0 h (s)je if(s) d ds e if(s) j (s)ids ) = = e i(f(S) f(0)) h (0)j (S)i h (S)j (0)i 1=2 exp ( i Z S 0 d ds f (s)ds Z S 0 h (s)j d ds j (s)ids ) = = e i ] ; (7)
and ] is indeed only a function of the path in projective Hilbert space. Incidentally, one sees from (7) that, taken separately, neither ' nor are projective quantities.

Secondly, ] is reparametrisation invariant, in the sense that under any strictly monotonical transformation s 7 ! , the expression (2) is invariant, as can be easily seen by noticing that d d = d ds ds d and d = ds d ds . We shall now recall what is perhaps the most natural geometric interpretation of the phase ]. This is provided by consideration of a connection (or, alternatively, of a parallel transport) on the Hopf bundle over the ray space. The Hopf bundle is the bre bundle over ray space whose bre over a ray consists of all unit vectors j i in the ray. A parallel transport of a unit vector eld j (s)i along a curve in projective Hilbert space is, roughly speaking, a rule that associates to any vector j (0)i a corresponding vector j (s)i in the bre over any other point of the curve. That is, it has to specify a phase for j (s)i. One such rule is provided by the condition2 h (s)j d ds j (s)i = 0:

(8)

If ( 8) is not satis ed, then j (s)i is picking up an additional phase during the (non-parallel) transport along the curve. In fact, if h'(s)j d ds j'(s)i = 0 and j (s)i = e i (s) j'(s)i, then

h (s)j d ds j (s)i = h'(s)je i (s) h i d (s) ds e i (s) j'(s)i + e i (s) d ds j'(s)i i = = i d (s) ds h'(s)j'(s)i + h'(s)j d ds j'(s)i = = i d ds (s); (9)
so that this additional phase is precisely given by = i R S 0 h (s)j d ds j (s)ids, that is (5). This justi es calling (5) a locally accumulated phase. Since in the de nition (2) of ] this phase is subtracted from the total phase (4), it is clear that ] is the phase that the vector j (s)i would have globally acquired if it had been parallel transported along the curve , and thus it is indeed a purely geometric phase. In fact, quite analogously to the case of Riemannian manifolds, this globally acquired geometric phase can be explained in terms of curvature of the ray space (see e. g. [START_REF] Page | Geometrical Description of Berry's Phase[END_REF]). This is more easily seen in the case of the AA phase, where is a closed curve, but is equally valid in the case of the AW phase, where one can close the curve by joining its endpoints through a geodesic (along which it can be shown that the geometric phase is zero), as in [START_REF] Samuel | General Setting for Berry's Phase[END_REF].

3 Non-Invariance of Geometric Phase Physical time t is not in general an appropriate parameter for a curve , since a mapping t 7 ! (j (t)i) need not be locally one-to-one. Nonetheless, it is easy to show that if (j (t)i) is constant over a certain time interval, then the phase given by ( 2) is zero for that time interval. Thus ( 2) is an expression for the geometric phase ] even when is expressed as a function of t:

e i ] = h (0)j (T )i h (T )j (0)i 1=2 exp ( Z T 0 h (t)j d dt j (t)idt ) ; (10)
(Only in this case maybe should the phase corresponding to (5) be called the dynamic phase.) Further, the expression (2) is independent of the choice of curve in Hilbert space also when t is substituted for s in (6); and it is invariant under time rescalings, i. e. under strictly monotonical transformations t 7 ! .

If is a function of time, we can consider the special case in which there is a curve t 7 ! j (t)i in Hilbert space projecting to that is given by the Schr odinger evolution of a state j (0)i with Hamiltonian H:

i h d dt j (t)i = Hj (t)i (11)
(where H will generally also be time-dependent). In this case, the geometric phase can be written as

e i ] = h (0)j (T )i h (T )j (0)i 1=2 exp ( i h Z T 0 h (t)jHj (t)idt ) : (12)
Time rescaling invariance is manifest under the transformation t 7 ! if one considers that the Hamiltonian, in order to preserve covariance of the Schr odinger equation, transforms as H = dt d H. And since the geometric phase is a function of the path alone, we can understand it as a universal property of the dynamics, in the sense that if the curve is induced by a Schr odinger evolution, it will in fact be independent of the particular Hamiltonian that drives the system along the path . As we have seen, this phase is further geometric in the strong sense of being determined (indeed being a measure of) the curvature of projective Hilbert space. Notice that the expression (12) is not itself projective, but only coincides with (2) if one chooses the one curve in Hilbert space projecting to which is also a solution to the Schr odinger equation with the given H. If one takes (6) to be an active transformation, it will change the actual curve in Hilbert space, and while d dt will still be perfectly well-de ned for that Hilbert-space curve, it will no longer be given by ( 11).

If the parameter t is physical time, one can enquire whether the geometric phase ] is invariant under a number of (passive) time-dependent unitary transformations of the Hilbert space, such as gauge and Galilei transformations. (It is trivial to show that ] is invariant under arbitrary time-independent unitary transformations.) The intuitive reason for not expecting geometric phases to be invariant in general under time-dependent unitary transformations is that the geometric phase is a function ] of a path in projective Hilbert space, but such a path is clearly not an invariant object under general time-dependent unitary transformations, just as a time-dependent trajectory in space is not invariant under general space{time transformations. Indeed, in constructing an invariant geometric phase, we shall be pursuing the analogy with the space{time path. Typically, under such a transformation U (t), a closed path : 0; T ] ! P(H) in projective Hilbert space will be mapped to an open path, since generally U (T ) 6 = U (0) and U (T ) (T ) 6 = U (0) (0). And even if we consider di erences of two open paths with the same endpoints, : 0; T ] ! P(H); : 0; S] ! P(H) proceeds along from (0) to (T ) = (T ) and back along to (0) = (0)), such a closed loop will be indeed mapped to a closed loop, but to a di erent one. Notice that along an open path not even the total phase is invariant. Instead for a di erence the total phase is invariant under arbitrary time-dependent unitary transformations, being directly related to the (observable) interference between the two paths and . 3 It is, indeed, only for such path di erences (as emphasised by Garc a de Polavieja (1997)) that the problem of the invariance of the geometric phase becomes interesting, or rather the problem of how to split invariantly the total phase into a geometric and a dynamic part.

Let us now look at speci c time-dependent transformations. The case of global gauge transformations is unproblematic. In fact, a global gauge transformation, j ~ (t)i := e iq (t)= h j (t)i ( 16) (where q is the charge of the particle), induces a simultaneous transformation of the type (6) on all curves t 2 0; T ] 7 ! (j (t)i; and we have seen that ( 2) is invariant under such transformations. In this case, also expression ( 12) is invariant, since under the (passive) transformation ( 16), the Hamiltonian transforms as H = e iq (t)= h He iq (t)= h q d (t) dt :

(17) This is because whenever we consider time-dependent unitary transforma-

tions U : j i 7 ! j ~ i = U j i, the Leibniz rule gives d dt j ~ (t)i = U d dt U 1 + dU dt U 1 j ~ (t)i: (18)
Thus, from ( 11) and ( 18), the Hamiltonian must transform as

H = U HU 1 + i h dU dt U 1 ; (19)
to ensure covariance of the Schr odinger equation. In particular, H transforms exactly as i h d dt , and one can use indi erently the representations ( 2) and ( 12) for the geometric phase. However, in the case of local gauge transformations and of Galilei transformations, the geometric phase is not invariant, as we can see as follows.

Take a (time-dependent) local gauge transformation as given by j ~ (t)i := e iq (Q;t)= h j (t)i:

(20)

From (18) one has that

e i ~ ] = h (0)je iq (Q;0)= h e iq (Q;T )= h j (T )i h (T )je iq (Q;T )= h e iq (Q;0)= h j (0)i ! 1=2 exp ( Z T 0 h (t)j d dt + iq h @ (Q; t) @t j (t)idt ) = = e i ] h (0)je iq (Q;0)= h e iq (Q;T )= h j (T )i h (0)j (T )i h (T )j (0)i h (T )je iq (Q;T )= h e iq (Q;0)= h j (0)i 1=2 exp ( iq h Z T 0 h (t)j @ (Q; t) @t j (t)idt ) : (21)
But now, while the middle factor on the right-hand side of ( 21) is a function of the endpoints of alone (in projective Hilbert space!), and so drops out if we consider closed loops, say , it is quite clear that R T 0 h (t)j @ (Q;t) @t j (t)idt is a function of the actual path, so that in general for di erences the last factor in (21) will be di erent from one. 4 The case of Galilei transformations has recently been treated by Sj oqvist, [START_REF] Sj Oqvist | Galileian Noninvariance of Geometric Phase[END_REF]. SBC, as we shall henceforth call them, consider a system with Hamiltonian H = (P q c A) 2 2m + V;

(22) with a vector potential A and a scalar potential V , both generally functions of Q and t, and where again q denotes the charge of the particle. SBC take a passive Galilei boost to be implemented by the unitary transformation5 j ~ (t)i := U(t)j (t)i := e iv(mQ Pt)= h = = e imv 2 t=2 h e imv Q= h e iv Pt= h : (23) By using ( 12), ( 19), ( 22), ( 23) and the identities U 1 QU = Q vt1;

4 Another way of seeing that the middle factor disappears is to notice that because of the projective nature of the geometric phase, the representations in Hilbert space of the two curves and can be chosen to have common endpoints. This amounts to choosing a single-valued representation in H of seen as a closed loop in P(H), so that the geometric phase can be computed using (3). For a single path , it is clear that by varying @ (Q;t) @t one can x the value of ] quite arbitrarily.

U 1 PU = P mv1;

(24) and U 1 dU dt = i h mv 2 2 v P ;

(25) they then calculate ~ ] as

e i ~ ] = e i ] h (0)je iv PT j (T )i h (0)j (T )i h (T )j (0)i h (T )je iv PT j (0)i ! 1=2 exp ( i h Z T 0 h (t)jv Pj (t)idt ) : (26) Using Ehrenfest's theorem, h (t)jPj (t)i = q c h (t)jAj (t)i + m d dt h (t)jQj (t)i; (27)
expression ( 26) becomes

e i ~ ] = e i ] h (0)je iv PT j (T )i h (0)j (T )i h (T )j (0)i h (T )je iv PT j (0)i ! 1=2 exp ( iq hc Z T 0 h (t)jv Aj (t)idt ) exp im h h h (T )jv Qj (T )i h (0)jv Qj (0)i i : (28)
But now again, one sees that while the second and fourth factor on the right-hand side of ( 28) are (projective) functions of the endpoints of only, and thus drop out when considering di erences , the factor containing R T 0 h (t)jv Aj (t)idt 6 = 0 depends on the actual path, so that in general it will be di erent from one for di erences . Thus, both for local gauge transformations and for Galilei transformations, geometric phases are generally not invariant, not even for closed loops. 6 6 These non-invariance results refute a claim by [START_REF] Kendrick | Invariance of the Geometrical Phase under Time-Dependent Unitary Transformations[END_REF] to the e ect that geometric phases would be invariant under arbitrary time-dependent unitary transformations U(t). Taking Galilei transformations to x the ideas, Kendrick in e ect chooses a frame in which geometric phases are de ned by means of (12), while in any other frame, they are given by a formula that essentially transforms back to the initial frame and explicilty depends on the transformation U(t), that is, on knowledge of one's state of motion with respect to the preferred frame! Thus, although it allows one to compute the same number in di erent frames, Kendrick's de nition of the geometric phase is plainly not invariant.

Previous Treatments of Non-Invariance

It was already seen very clearly by [START_REF] Aharonov | Phase Change during a Cyclic Quantum Evolution[END_REF] that geometric phases need not be invariant under time-dependent unitary transformations. In that paper and, in an improved form (which we shall more or less follow here), in [START_REF] Anandan | Geometric Quantum Phase and Angles[END_REF] they put forth a gaugeinvariant de nition of geometric phase, which is formally very similar to, if di erent in spirit from, the one we shall present below. [START_REF] Anandan | Geometric Quantum Phase and Angles[END_REF] de ne a gauge-invariant state as:

j ^ (t)i := exp iq h Z t 1 A 0 (t 0 )dt 0 j (t)i; (29)
where A 0 (t 0 ) is the electric potential. They then apply the usual de nition of the geometric phase (here we take (2)) to the gauge-invariant state j ^ (t)i.

Call the result e i ] :

e i ] := h ^ (0)j ^ (T )i h ^ (T )j ^ (0)i ! 1=2 exp ( Z T 0 h ^ (t)j d dt j ^ (t)idt ) : (30)
This is manifestly gauge-invariant. AA apply their formula only to paths that are cyclic in the speci c sense that j ^ (T )i = e i j ^ (0)i, but the above formula clearly provides a gauge-invariant de nition also in the non-cyclic case. If j (t)i satis es the Schr odinger equation

i h d dt j (t)i = " (P q c A) 2 2m + qA 0 # j (t)i; (31) then j ^ (t)i satis es i h d dt j ^ (t)i = (P q c A) 2 2m j ^ (t)i; ( 32
)
where H k :

= (P q c A) 2 2m
is the kinetic part of the Hamiltonian, so that (30) can also be written as

e i ] = h ^ (0)j ^ (T )i h ^ (T )j ^ (0)i ! 1=2 exp ( i h Z T 0 h ^ (t)jH k j ^ (t)idt ) : (33)
We shall construct our de nition below to be numerically equivalent to (30) for closed loops (as those considered by AA), but with a quite different interpretation. Jeeva Anandan (1997) informs me of the possibility of basing a Galilei-invariant geometric phase on the de nition of a Galileiinvariant state (analogous to (29)). However, this is no longer equivalent to the proposal below.

In an article prompted by SBC (1997), Garc a de Polavieja (1997) proposes to consider only phase di erences and to choose a special gauge in which v A = 0 (for given v: it is always possible to choose a gauge such that a particular component of the vector potential vanishes). This will make the third factor on the right-hand side of (28) vanish, so that the geometric phase around a closed loop will be, indeed, invariant. A subtlety in Garc a de Polavieja's treatment is that since the derivation of (28) assumes Ehrenfest's theorem (27), the vector potential A appearing in (28) must in fact be the one appearing in the Hamiltonian that drives the system along the path . If however and are both de ned on 0; T ], then there is no guarantee that v A = 0 may be gauged away along both paths. Instead, Garc a de Polavieja considers the di erence of two successive paths : 0; T ] ! P(H) and : T; T 0 ] ! P(H) with common endpoints (0) = (T ) and (T ) = (T 0 ), and such that (T ) = (T ) (so that, in fact, the closed path is a double loop in projective Hilbert space). In this case, the geometric phase along in the given gauge will be indeed invariant under all Galilei boosts in the direction of v.

5 The Extended Geometric Phase

In this section and the next, we shall de ne a splitting of the total phase into two parts, analogous to the geometric phase and the locally accumulated phase, that is indeed gauge-and Galilei-invariant. We shall then see in Section 7 how this `geometric' phase can further be associated with a notion of parallel transport on a parametrised projective Hilbert space P(H) t , by which I mean a bre bundle over the time axis with bres isomorphic to P(H). Then in Section 8 we shall sketch the classical analogue in Hamiltonian mechanics. Finally, in Section 9, we shall apply our newly de ned phase to [START_REF] Berry | Quantum Phase Factors Accompanying Adiabatic Changes[END_REF] calculation of the Aharonov{Bohm e ect for a box transported around a solenoid.

The easiest way to introduce our phase is to start with AA's proposal (30). The mapping (29) from j (t)i to j ^ (t)i has the form of a local gauge transformation. (It is the transformation to a gauge in which the electric potential vanishes.) Applying the transformation formula ( 21) for the geometric phase under local gauge transformations, we can write (30) in terms of the original states j (t)i as:

e i ] = h (0)j (T )i h (T )j (0)i 1=2 exp ( Z T 0 h (t)j d dt + iq h A 0 j (t)idt ) : (34)
For a closed loop we can write, analogously to (3):

] = i I h (t)j d dt + iq h A 0 j (t)idt = = i Z T 0 h (t)j d dt + iq h A 0 j (t)idt +i Z T 0 h'(t)j d dt + ie h A 0 j'(t)idt (35)
where j (t)i and j'(t)i are representations of and which agree at t = 0 and t = T. We shall presently check that (34), although not gauge-invariant for open paths (unlike (30)), is gauge-invariant for closed loops . It is clear that in this case it is numerically equal to AA's gauge-invariant phase.

In the special case in which the Hamiltonian is given by H = (P q c A) 2 2m + qA 0 ;

(36) the phase (34) becomes

e i ] = h (0)j (T )i h (T )j (0)i 1=2 exp ( Z T 0 h (t)jH k j (t)idt ) ; (37)
where H k is again the kinetic part of the Hamiltonian (36). The phase in the second factor on the right-hand side of (37), call it ], may be aptly called the kinetic phase rather than the dynamic phase.

The phase (34) di ers from (2) by a factor which is the exponential of the integral of an expectation value. Since expectation values are projective quantities, (37) is thus also projective, in the sense that it does not depend on the particular curve t 7 ! j (t)i in Hilbert space used to de ne it. It is further time rescaling invariant, since it is clear that in order to maintain covariance of the Schr odinger equation A 0 transforms under t 7 ! to Ã0 = dt d A 0 .

Notice that, despite being independent of the curve in Hilbert space and of the choice of the time parameter, and thus being a function of the path , the phase ], unlike ], is not a function of alone. In fact, it need not be zero for a time interval in which (j (t)i) is constant. This precludes an interpretation in terms of the geometry of projective Hilbert space P(H) but, as we shall discuss in Section 7, it is readily understandable in terms of the analogous geometric structures on the parametrised ray space P(H) t . Also, the phase ] truly depends on the Hamiltonian that drives the system along the path (and requires the existence of a Hamiltonian of the correct form (36), or (38) below). In this sense, ] is no longer a universal aspect of the dynamics. 7 If the Hamiltonian contains a further scalar potential, say V 0 , besides the electric potential A 0 , H = (P q c A) 2 2m + V;

(38) with V = qA 0 + V 0 , it is also quite natural to use the full potential V in the de nition (34), so that it now reads e i ] := h (0)j (T )i h (T )j (0

)i 1=2 exp ( Z T 0 h (t)j d dt + i h V j (t)idt ) ; (39) or ] = i I h (t)j d dt + i h V j (t)idt (40)
for a closed loop. If V 0 6 = 0, this ensures that (37) is valid. We shall see that the phase ] shares many of the invariance properties that hold for ].

7 Notice that it is only required that j (t)i and j'(t)i be any representations in H of paths in P(H) that arise through a Schr odinger evolution. Because ] is projective, the actual representations of and need not (both) be solutions of the respective Schr odinger equations.

Finally, one should point out that if one de nes a phase ] analogously to ], but with the Hamiltonian H in place of the electric potential A 0 , e i ] := h (0)j (T )i h (T )j ( 0

Invariance Properties

We shall now derive the invariance properties of ]. Namely, we shall show that di erences ], where and have common endpoints in ray space, are invariant under all gauge transformations and Galilei boosts and under transformations to linearly accelerated coordinate systems.

Let a (global or) local gauge transformation be given by j ~ (t)i := U(t)j (t)i := e iq (Q;t)= h j (t)i:

(42) By ( 18) and ( 42), one has

d dt = U d dt + iq h d (Q; t) dt : (43)
Further, it is well-known that under (42) the Hamiltonian ( 22) transforms as H = (P qr (Q; t) q c A) 2 2m + V q d (Q; t) dt (44) (in fact, this follows from ( 19)), so that in particular Ṽ = V q d (Q; t) dt :

(45)

If V = qA 0 + V 0 we can write Ã0 = A 0 d (Q; t) dt (46)
and Ṽ0 = V 0 :

(47) In fact, ( 46) is the standard transformation behaviour for the electric potential, and it is indeed obvious that d (Q;t) dt must be part of the electric potential, since in (45) the particle couples to it through its charge q.

From ( 43) and ( 45) it follows that

d dt + iq h Ã0 = U d dt U 1 + iq h d (Q; t) dt + iq h A 0 iq h d (Q; t) dt = = U d dt U 1 + iq h A 0 = = U d dt + iq h A 0 U 1 (48)
(since A 0 is a function of Q and thus, by ( 42), commutes with U).

Inserting ( 48) into (39) we nally obtain e i ~ ] = e i ] h (0)je iq (Q;0)= h e iq (Q;T )= h j (t)i h (0)j (T )i h (T )je iq (Q;T )= h e iq (Q;0)= h j (0)i h (T )j (0)i

! 1=2 ; ( 49
)
where the second factor on the right-hand side is projective and a function only of the endpoints of , so that for two curves in ray space and with common endpoints one has e i ~ ~ ] = e i ] :

(50)

Because of (45), V can be substituted for qA 0 throughout (48), and thus the analogous invariance under gauge transformations holds also for ].

In order to prove that ] is invariant also under Galilei boosts, we note that under a Galilei boost (23), a Hamiltonian of the form ( 22) transforms as

H = (P q c UAU 1 ) 2 2m + U(V q c v A)U 1 ; (51)
as can be shown for instance using ( 19), ( 24) and ( 25) (for details see [START_REF] Takagi | Quantum Dynamics and Non-Inertial Frames of Reference. I[END_REF] and [START_REF] Brown | The Galilean Covariance of Quantum Mechanics in the Case of External Fields[END_REF]). In particular, the scalar potential V transforms as Ṽ = U(V q c v A)U 1 :

(52)

Again, since the coupling constant appearing in q c Uv AU 1 is proportional to the charge q, this can be recognised as part of the electric potential with respect to the new frame, so we can write also

Ã0 = U(A 0 q c v A)U 1 (53)
and Ṽ0 = UV 0 U 1 :

(54)

Given that, comparing ( 2) and ( 39),

e i ] = e i ] exp ( iq h Z T 0 h (t)jA 0 j (t)idt ) ; (55)
the transformation formula ( 28) for ] yields: e i ~ ] = e i ] h (0)je iv PT j (T )i h (0)j (T )i h (T )j (0)i h (T )je iv PT j (0)i

! 1=2 exp ( iq hc Z T 0 h (t)jv Aj (t)idt ) exp im h h h (T )jv Qj (T )i h (0)jv Qj (0)i i exp ( iq h Z T 0 h ~ (t)j Ã0 j ~ (t)idt ) ; ( 56
)
where now j ~ (t)i is given by ( 23).

From (53) we thus have e i ~ ] = e i ] h (0)je iv PT j (T )i h (0)j (T )i h (T )j (0)i h (T )je iv PT j (0)i

e i ~ ] = e i ] h (0)je iv PT j (T )i h (0)j (T )i h (T )j (0)i h (T )je iv PT j (0)i ! 1=2 exp im h h h (T )jv Qj (T )i h (0)jv Qj (0)i i : (58)
Again, given ( 52), V can be substituted throughout for qA 0 . As before, the The invariant behaviour of ] generalises further to transformations to linearly accelerating frames, as can be seen as follows. First of all, the unitary operator implementing a transformation to a linearly accelerated frame is given by U(t) := e im R t 0 v(u) 2 du=2 h e imv(t) Q= h e ir(t) P= h (60) (compare ( 23)), where r(t) and v(t) are the position and velocity of the origin of the moving frame, respectively (see Takagi (1991, x3)).

SBC's (1997) derivation of the transformation behaviour ( 28) of ] is virtually unaltered. It is enough to substitute a time-dependent velocity v(t). One has then: e i ~ ] = e i ] h (0)je ir(T ) P j (T )i h (0)j (T )i h (T )j (0)i h (T )je ir(T ) P j (0)i

! 1=2 exp ( i h Z T 0 h (t)jv(t) Pj (t)idt ) : (61)
Using again ( 27), one obtains e i ~ ] = e i ] h (0)je ir(T ) P j (T )i h (0)j (T )i h (T )j (0)i h (T )je ir(T ) P j (0)i

! 1=2 exp ( iq hc Z T 0 h (t)jv(t) Aj (t)idt ) exp im h h h (T )jv(T ) Qj (T )i h (0)jv(0) Qj (0)i i : (62)
The transformation behaviour of the Hamiltonian ( 22) under ( 60) is no longer given by ( 51), not even with a time-dependent v(t), but by H = (P q c UAU 1 ) 2 2m + U V q c v(t) A U 1 a(t) Q; (63) where a(t) is the generally non-zero acceleration of the moving frame (see again Takagi (1991, x3)). The additional potential a(t)Q is an inertial potential, which belongs to Ṽ0 : Ṽ0 = UV 0 U 1 a(t)Q;

(64) while for the electric potential we still have the transformation behaviour ã0 = U A 0 q c v(t) A U 1 :

(65)

Thus the invariance result for ] generalises also to the case of linearly accelerated frames. Instead, in the transformation formula for ] there appears a new term,

exp ( i h Z T 0 h (t)ja(t) Qj (t)idt ) ; ( 66
)
which does not depend only on the endpoints of , thus destroying in general the invariance of the phase di erences.

Finally, we show that the representation (41) of the total phase for a path di erence is invariant under arbitrary time-dependent unitary transformations U (t), as indeed it should.

Since H = U HU 1 + i h dU dt U 1
(67) and

d dt = U d dt U 1 + dU dt U 1
(68) (recall ( 18) and ( 19)), we have that

d dt + i h H = U d dt + i h H U 1 : (69)
And thus, by the de nition (41) of ],

e i ~ ] = e i ] h (0)jU 1 (0)U(T )j (T )i h (0)j (T )i h (T )jU 1 (T )U(0)j (0)i h (T )j (0)i ! 1=2 :
(70) Since the second factor on the right-hand side depends, yet again, only on the endpoints of in ray space, di erences ] ] = ] (where and have common endpoints) are invariant under arbitrary time-dependent unitary transformations U (t).

Geometric Interpretation

In the case in which j (t)i is a solution of a Schr odinger equation, as we have seen, ] is also given by

e i ] = h (0)j (T )i h (T )j (0)i 1=2 exp ( i h Z T 0 h (t)jHj (t)idt ) ; (71)
so that subtracting the dynamic phase from the total phase, one indeed obtains a phase that is projective, independent of the particular Hamiltonian inducing the evolution, and geometric in the strong sense that it is given by the curvature of the projective Hilbert space.

In the case of the phase ] introduced in Section 5, we have noted already that it is also projective and independent of time rescalings, but that it does not depend only on the path in ray space P(H). We shall now consider ] instead in relation to a path, again denoted by , in what we might call time{ray space, that is the parametrised ray space P(H) t , a space in which the path becomes an invariant object, much like a worldline in relativistic space{time. The phase ] will still not be independent of the Hamiltonian that drives the system along the path, but it will be interpretable as arising from a Hamiltonian-dependent connection on time{ ray space.

In order to obtain a geometric characterisation of ] we have to introduce on time{ray space a structure analogous to parallel transport in the Hopf bundle. We rst construct such a bre bundle, by de ning the bre over a point ( ; t) 2 P(H) t as the set of unit vectors in the ray : n j i j j j j = 1;

(j i) = o : (72)
A curve in time{ray space will now be given by a locally one-to-one mapping : s 2 0; S] 7 ! (j (s)i); t(s) :

(73) And a parallel transport in the bundle will clearly be given by a rule that xes the phase of j (s)i.

The rule of parallel transport that yields ] is in fact h (s)j d ds j (s)i + iq h h (s)j dt(s) ds A 0 (t(s))j (s)i = 0

(or analogously for ] with V (t(s)) in the place of qA 0 (t(s))). It is clear, by analogy to (9), that this provides us with a condition on the phase of j (s)i. We can thus use it to de ne a locally accumulated phase, as in (5), and a phase

e i ] := h (0)j (S)i h (S)j (0)i 1=2 exp ( Z S 0 h (s)j d ds + iq h dt ds A 0 j (s)ids ) ; (75)
again denoted by ], which is manifestly projective and reparametrisation invariant. In fact, for the case in which dt ds = 1, this is identical to the definition (39) of ]. Indeed, it is identical to (39) in the more general case in which dt ds 6 = 0 almost everywhere along the curve, which corresponds to a time rescaling t 7 ! = s (and possibly inverting the direction of time, as when going backwards along ). Time rescaling invariance is explicitly built into the de nition of (75) as a special case of reparametrisation invariance, while it is implicit in the de nition (39). In fact, the transformation behaviour of A 0 is explicitly built into (75). Thus, (75) gives us an explicit interpretation of our phase ] as a geometric phase de ned in terms of a parallel transport on time{ray space. It becomes identical to (2) in the case in which dt ds = 0 all along the curve, so that ] can be interpreted as a special case of ] for the case of parallel transport over time{ray space at any xed time, which (as it should) coincides with parallel transport over ray space.

Classical Analogue

The geometric phase (2) has a classical analogue, as shown by [START_REF] Anandan | Non Local Aspects of Quantum Phases[END_REF], in the canonical invariant S 0 = I pdq (76) over a closed loop. In fact, this is the classical limit of the geometric phase for the case of coherent states. 8 To be precise, the analogue of ( 2) for open paths has been given by Sj oqvist and Hedstr om (1997) as 1 2 Z pdq qdp:

(77)

If we are interested only in loops of the form , however, we need to consider only (76).

8 Notice also the analogy with the adiabatic invariant I = 1 2 I p dq (Landau and Lifshitz, 1976, Section 49)

check].
The quantity ( 76) is invariant under arbitrary canonical transformations, as long as these do not depend on time. In the more general case, one usually takes instead the quantity S = I pdq Hdt; (78) where H is the Hamiltonian function of the system. This quantity is now invariant under arbitrary time-dependent canonical transformations. (In fact, ( 78) is the canonical invariant in the extended phase space, i. e. the phase space with the additional pair of conjugate variables q 0 := t and p 0 := H.) If ( 76) is the analogue of (2), it is easy to recognise (78) as the analogue of (41), i. e. of the total phase, which as we have seen in Section 6 is invariant under arbitrary time-dependent unitary transformations.

We can now ask whether the extended geometric phase ] de ned in (34) or (39) also has a classical analogue, which is invariant under canonical transformations implementing local gauge and Galilei transformations. And indeed, it has. We shall now illustrate the case of Galilei transformations.

Let the Hamiltonian function of the system be H = (p q c A) 2 2m + V;

(79) where V = qA 0 + V 0 , and consider a Galilei transformation q ! q := q vt; (80) p ! p := p mv: (81) This transformation has a generating function, which can be written, say in terms of the variables q and p, as (q; p; t) = p(q vt) + mvq:

(82)

One then has, indeed (Landau and Lifshitz, 1976, Section 45), p i = @ @q i ; qi = @ @p i ; H = H + @ @t :

(83)

The last of these equations describes how the Hamiltonian function has to transform in order for the Hamilton equations to be covariant, i. e. satis ed also with respect to the new coordinates. In this case, the Hamiltonian function transforms as: H = H pv:

(84) Thus, from ( 79), ( 80) and ( 81),

H = (p + mv q c A) 2 2m + V = = (p q c A) 2 2m + pv + 1 2 mv 2 q c vA + V (85)
(where both A and V are understood as functions of q + vt), and, from this and ( 84), H = (p q c A) 2 2m + V q c vA + 1 2 mv 2 :

(86) The last term is a rescaling of energy, which can be safely ignored, while we see that the vector and scalar potentials transform as Ã(q) = A(q); ( 87) Ṽ (q) = V (q) q c vA(q); Now, if and are solutions to the Hamilton equations, in particular with H given by ( 79), then dq = _ qdt and further p = m _ q + q c A: (95) so that ( 76) is, indeed, not invariant in general under Galilei transformations.

From this and from ( 88) and ( 90), however, it is clear that the quantities are both invariant under the Galilei transformation (80){(81). We have thus found phase space loop integrals distinct from the canonical invariant (78) which are nevertheless also invariant under Galilei transformations. (As can be readily seen from the analogous calculations, the results for gauge transformations and for transformations to linearly accelerated frames are also valid in the classical case.)

9 Aharonov{Bohm E ect Consideration of the classical analogue shows that, even if the geometric phase (2) is not (gauge-nor) Galilei-invariant, it surely has an important theoretical role to play, just as the canonical invariant (76). Non-invariance under gauge or Galilei transformations, however, raises the question of the observability of the geometric phase, as discussed by SBC (1997). In their discussion, SBC note how all experiments to date seem either to consider only the spin degrees of freedom | which makes them independent of spatially dependent transformations like gauge or Galilei transformations | or to consider new quantities which are analogues of the geometric phase for spin degrees of freedom (e. g. in describing interferometry experiments) | which has the same e ect.

There is, however, in the literature a thought experiment that does, indeed, involve the geometric phase associated with the spatial degrees of freedom of a system, and in which the gauge-and frame-dependence will be manifested. This is Berry's derivation of the Aharonov{Bohm e ect as the di erence in geometric phases between a particle in a box at rest and a particle in a box which is transported around a solenoid and brought to coincide again with the rst box. I wish to point out, as is evident already from Berry's treatment, that this calculation and thus the identi cation of the AB phase as a geometric phase depend on the chosen gauge (and frame). Instead, applying the de nition of the extended geometric phase ] to this obviously time-dependent problem yields an identi cation of the AB e ect as geometric which is independent of gauge or frame.

Let us recall [START_REF] Berry | Quantum Phase Factors Accompanying Adiabatic Changes[END_REF] calculation, recast slightly in terms of what we have discussed above. Berry considers a box, or rather two boxes, one at rest and one transported from the same location as the rst around a ux line and back, and such that the two boxes are never interpenetrated by the ux line. (The state of a particle will have to be localised within a box, so the potential walls are in nite.) We can write the Hamiltonian for the particle in such a box in the form

H = (P q c A(Q)) 2 2m + V (Q R(t)); (98)
where R(t) = const for the rst box and equals a cyclic path in R-space for the second box.

As Berry points out, the eigenvalues E n of H do not depend on R, nor on the presence or absence of the vector potential A (since the state of the particle is actually inside the box, the latter only induces an additional Dirac phase factor in the solution). Thus, denoting by jn(R)i the eigenstate of H (depending on R) corresponding to E n , we have: Hjn(R)i = E n jn(R)i; denoting the ux of the magnetic eld. Thus, the geometric phase is shown to be equal to the AB phase, and is, indeed, independent of n, so that the result is true whatever the state of the particle in the box.

(R)jrn(R)i = = Z Z Z d 3 r n (r R) iq h A(R) n (r R) + r R n (r R) = = iq h A(R); (102
One can check explicitly that Berry's de nition ( 101) is the same as (3) above, if one takes j (t)i = jn(R(t))i, which in fact solves the Schr odinger equation with Hamiltonian (98) | despite the fact that Berry's de nition is based on the notion of a path in the parameter space of the Hamiltonian (i. e. R-space), while AA's de nition which we are using is based on the notion of a path in projective Hilbert space. (As a matter of fact, as pointed out already by Berry, there is no need of adiabatic approximation in this case: j (t)i is a solution of the Schr odinger equation irrespective of how fast the box is being moved.) Thus, since Berry's C] is equal to ] in the sense of AA (denoting by and the paths in projective Hilbert space of the two boxes, respectively), the result (104) depends both on the chosen gauge, as we have discussed quite in general in Section 3 and as just remarked above, and on the chosen frame, as follows from SBC's (1997) discussion, again as summarised in Section 3. For instance, one can generally nd a Galilei transformation such that C] = 0, so that the AB phase becomes entirely dynamic! Such a case can now be prevented by use of the extended geometric phase . In fact we have that for this case: where we have used the fact that the energy of a particle stemming from the electric potential in either box (if any) is independent of R(t). (The same is true of ] and V .)

Thus, we have shown that our gauge and Galilei-invariant phase ] is also equal to the AB phase in the case considered by Berry, thereby giving a gauge-and Galilei-invariant identi cation of the AB phase with a geometric-type phase.

Conclusion

My investigation was motivated by the quest for an invariant de nition of geometric phase, and was carried out pursuing the analogy with space{time concepts. As to this second point, I have introduced the so-called time{ ray space, where one could consider paths analogous to paths in space{ time. I believe such a construction has independent interest. As to the rst point, the phase I have constructed above | although retaining many of the geometric features of the AA phase | has lost the independence of the Hamiltonian. My de nition of a connection in time{ray space depends itself on the Hamiltonian that drives the system along the path. This may be a disturbing feature of the phase (although there may be some interest in considering the connection as thus dependent on the dynamics, which could be an interaction dynamics in the case of , say, a Born{Oppenheimer framework). On the other hand, the connection introduced on time{ray space may be related to the electromagnetic connection on space{time. Finally, the extended geometric phase is a phase quantity distinct from the total phase, which however like the total phase is invariant under gauge and Galilei transformations. The phase thus de ned may be not geometric in the same sense as the original geometric phase, but its invariant behaviour makes it a real physical quantity.

  boosts and any closed loops enclosed by curves and in projective Hilbert space. (The same is true of ].)

  is over a closed loop in extended phase space, the last two terms do not contribute and we obtain the nal result:

C

  along the closed loop (in R-space) given by the di erence of the paths for the two boxes is calculated by Berry as

  ) assuming that n is normalised and real-valued. In fact,0 = r R ( n n ) = 2Re n r R n ;(103) so the second term in the triple integral in (102) vanishes if n r R n has no imaginary part. This can always be achieved, but clearly depends on the choice of gauge. From (101) and (102) one has then:

This is the so-called real geometric phase. AW treat extensively also the complex geometric phase, which we do not discuss here.

If the bre is not restricted to the unit vectors, one obtains the line bundle over projective Hilbert space. In this case, the real part of (8) becomes a non-trivial requirement that the norm of j (s)i be constant during parallel transport. Considering the line bundle is more in tune with de nition (1) as given by AW.

A projective expression for the total phase will be given below in Section 5.

See e. g.Fonda and Ghirardi (1970, x2.5). Here, as below in (60), the rest frame and the moving frame coincide at t = t 0 = 0 (standard con guration). Finally, notice that for operators A and B that commute with their commutator, the identity e A+B = e A e B e A;B]=2 holds.

Acknowledgements

It is a pleasure to thank my friends and colleagues Henrik Carlsen, Joy Christian, Konstyantin Ilienko and in particular Jeeva Anandan, Harvey Brown, Gonzalo Garc a de Polavieja and Erik Sj oqvist for useful, stimulating and often invaluable discussions, suggestions and criticism. I have written this paper as a British Academy Postdoctoral Research Fellow at Balliol College, Oxford.


Elise Crull
email: ecrull@nd.edu

Guido Bacciagaluppi
email: g.bacciagaluppi@abdn.ac.uk

Translation of: W. Heisenberg, 'Ist eine deterministische Ergänzung der Quantenmechanik möglich?'

Introduction

Immediately following the publication of the EPR paper in the spring of 1935, Pauli wrote to Heisenberg suggesting he should develop his own response to it. Heisenberg took up Pauli's suggestion, and on 2 July, he sent Pauli a draft entitled: 'Ist eine deterministische Ergänzung der Quantenmechanik möglich?' 1 At the end of August, Heisenberg also sent a carbon copy of a typescript to Bohr, asking for comments, and mentioning he was thinking of sending it to Die Naturwissenschaften for publication. 2 Bohr was puzzled by certain aspects of the argumentation. Although Heisenberg clarified these aspects in a subsequent letter, he postponed the decision to 1 publish until further discussion with Bohr in October in Copenhagen. 3 As it happens, the paper was never published during Heisenberg's lifetime. A not quite faultless transcription was published in volume 2 of Pauli's scientific correspondence (Pauli 1985, pp. 409-418). The original manuscript is in Heisenberg's Nachlaß in the Werner-Heisenberg-Archiv in Munich, and is reproduced in the Archive for the History of Quantum Physics (microfilm 45, section 11).

More than a direct reply to the EPR paper, Heisenberg's draft is a presentation of Heisenberg's own views about the completeness of quantum mechanics. In particular, the draft contains what appears to be Heisenberg's fullest presentation of his argument about the movable 'cut' between what is counted as part of the system to be observed and what is counted as part of the means of observation (this constitutes the bulk of Heisenberg's §1). It also contains (at the beginning of §2) what appears to be the earliest distinction between what are now called 'contextual' and 'non-contextual' hidden variables, even though Heisenberg quickly dismisses the contextual case as irrelevant.

The crux of Heisenberg's argument is presented in the remainder of his §2. He argues that the only place where additional variables could supplement the quantum mechanical description is at the location of the cut, say between the system A to be measured and a measuring device B. The cut, being movable, could also be placed between the combined system A + B and a further system C. In this case A + B can be given an entirely quantum mechanical description. As a consequence, system B might be used to measure a variable that is complementary to the one originally considered (say, momentum rather than position). It was this particular step in the argument that was most unclear to Bohr. If we grant the point, however, the assumed supplementation of the quantum mechanical description by way of additional variables (say, position) turns out to be incompatible, according to Heisenberg, with the quantum mechanical predictions for the measurement of the complementary variable. The latter point is illustrated with 3 Heisenberg to Bohr, 28 August 1935, Bohr to Heisenberg, 15 September 1935, Heisenberg to Bohr, 29 September 1935, Bohr to Heisenberg, 1 October 1935 (AHQP-BSC, microfilm 20, section 2), and Heisenberg to Bohr, 5 October 1935 (BSCSupp-HEI-351005t) (Heisenberg's letters in German, Bohr's letters in Danish). In a further letter to Margarethe Bohr dated 13 September 1935 (BSCSupp-HEI-350913t, in German), Heisenberg states explicitly that he will not send off the manuscript until he and Bohr have agreed on the content.

an example used by Heisenberg already in 1927, where he argues that the assumption of particle trajectories in a diffraction experiment on a grating is incompatible with the observed quantum mechanical interference. 4Heisenberg's final §3 summarises the preceding argument by reference to Grete Hermann's regrettably little-known essay 'Die naturphilosophischen Grundlagen der Quantenmechanik' (Hermann 1935a). Indeed, one can recognise the similarity with some of Hermann's main views in most of the section (except for the wavefunction example used by Heisenberg). 5We have given an analysis of Heisenberg's draft in a recent paper [START_REF] Bacciagaluppi | Heisenberg (and Schrödinger, and Pauli) on hidden variables[END_REF]. Some aspects of Heisenberg's draft are further clarified by the correspondence with Bohr, in particular Heisenberg's letter to Bohr of 29 September 1935, which clarifies in what sense system B can be used to measure two complementary variables. What was referred to implicitly here was the Heisenberg microscope, in which the electron A interacts with the photon B, which is then chosen to be observed either in the image plane (yielding a measurement of position) or the focal plane of the microscope C (yielding a measurement of momentum). Making the reference to the Heisenberg microscope explicit further emphasises the parallels between Heisenberg's and Hermann's views.

The present translation is based on Heisenberg's manuscript and corrects the mistakes in the published transcription of 1985. We have footnoted all discrepancies (giving first the manuscript reading, then the transcription), except when they constitute merely differences in spelling or punctuation with no implications for the meaning of the sentence. We include in our bibliography below also the complete version of the references given by Heisenberg in the original footnotes.

For permission to translate Heisenberg's manuscript, we wish to thank most warmly Helmut Rechenberg of the Werner-Heisenberg-Archiv in Mu-Schrödinger, E. (1934). Über die Unanwendbarkeit der Geometrie im Kleinen. Die Naturwissenschaften, 22(31), 518-520.

Seevinck, M. (no date). Translation into English of G. Hermann (1935a) W. Heisenberg: Is a deterministic completion of quantum mechanics possible?

The laws of quantum mechanics contain in many cases statements about the probability for a definite event taking place. Although this statistical character of quantum mechanics is intimately connected to the general features of the description of nature investigated by Bohr 6 that have come to light in atomic physics and are described by the concept of 'complementarity', yet, in discussions on 7 atomic theory the question is raised again and again whether quantum mechanics might not be amended later to a deterministic theory through new physical results. 8 It is natural to consider quantum mechanics at first to be an incomplete description of nature; for it seems to follow from entirely general principles that wherever an exact prediction of what happens in the future has not succeeded, this failure should be seen as signalling the presence of a problem that is yet unsolved and will need to be solved in the future. An argument of this kind, in light of the experimental success of quantum mechanics, assumes in general that quantum mechanics gives a correct description of nature. However, it ties this assumption to the hope that later research will bring to light a hitherto hidden network of causal connections behind the statistical connections of quantum mechanics -rather like the way classical mechanics is hidden behind the temperature and entropy concepts of thermodynamics. These causal connections need not refer at all to anschaulich classical variables of physical systems. Rather, it is inferred from the validity of the indeterminacy relations that classical concepts allow no adequate description of atomic phenomena, that therefore one needs to develop new concepts to be perhaps associated with hitherto unknown physical properties of atomic systems. For instance, apart from the physical properties that are exhaustively specified by determining its stationary state, the nucleus of the radium atom could, say, possess still other hitherto unknown physical properties, knowledge of which would make possible an exact prediction of the time of radioactive decay. The following considerations are meant to show under very general assumptions that such a deterministic completion of quantum mechanics is impossible, thus that one might maintain the hope for a description of nature that is determinis-6 N. Bohr, Solvay Conference 1927;Naturwiss. 16, 245 (1928); Atomtheorie und Naturbeschreibung. Berlin 1931; Faraday-lecture, Journ. of the Chem. Soc. 1932. p. 349;Lys og liv, Natur. Verden 17, 49 (193[3]). Phys. Rev. Forthcoming [presumably (1935)].

7 [über/um] 8 E.g., M. v. Laue, Naturw. 20, 115 (1932);Naturwiss. 22, 439 (1934);E. Schrödinger, Naturwiss. 22, 518 (1934); A. Einstein [et al.], Phys. Rev. 47, 777 (1935).

tic in the traditional sense only if one chose to consider the most important experimental successes of quantum mechanics to be accidental. The essential content of the following trains of thought is already to be found in the earliest discussions of the fundamental interpretation of quantum mechanics; 9 presenting it anew is perhaps nonetheless justified by the criticism of quantum mechanics repeatedly expressed in the most recent literature. §1. Quantum mechanics represents a physical system by a wavefunction in a configuration space whose number of dimensions is determined by the number of degrees of freedom of the system in question. The square of the absolute value of the wavefunction at a specific point of this space gives the probability that the anschaulich physical quantities denoted by the coordinates of the space take on the specific values corresponding to that point, if the system is observed with regard to these values. The formalism of quantum mechanics is thus based on the assumption that a physical system can be represented by classical-anschaulich variables, and that, as in classical theory, there can be an objective sense independent of the processes of observation in speaking of the actual value of a specific physical quantity, e.g. of the 'position of the electron'.

The quantum mechanical way of describing nature thus begins with a peculiar rift: on the one hand, one proceeds from the assumption that the task of physics is the description and synthesis in terms of laws of anschaulich,objective processes in space and time; on the other hand, one uses for the mathematical description of physical processes these wavefunctions 10 in multi-dimensional configuration spaces, which in no way can be seen simply as representatives of the objective happening in space and time, as can, say, the coordinates of a point-mass in classical mechanics. This rift manifests itself in an arbitrariness in the application of quantum mechanics: should only the atomic system to be observed be represented by a wavefunction, and the devices used for its 11 observation be treated according to the laws of classical physics, or should also the devices be represented by wavefunctions according to the laws of quantum mechanics, where in the end only the observation of the measuring device, say, the observation of 9 N. Bohr, l.c.; W. Heisenberg, Z. Phys. 43, 172 (1927). Cf. also W. The answer to this question is: the quantum mechanical predictions about the outcome of an arbitrary experiment are independent of the location of the cut just 12 discussed. Since this theorem is of crucial importance for the internal coherence [für den inneren Zusammenhang] of quantum mechanics, its 13 proof shall be given here in detail. Let an atomic system A be given, from which information reaches the observer by means of the measuring devices B, C, etc. The systems B, C... should be capable of being regarded as measuring devices in the sense of classical physics, i.e. from a specific reaction, say, of system C, it should be possible to infer univocally to a reaction of system B, and from 14 this to a specific behaviour of A.

Atomic System A → B → C → ... → Observer.
For this to be possible, a certain physical interaction between the systems A, B, C, etc., must be present. For example, it could be that the interaction energy between A and B is non-zero only when the coordinates q A of system A possess the quite specific values q A . From the occurrence of a reaction in B one could then infer that the coordinates of A have in fact assumed these particular values q A . Let us make this special assumption about the interaction energy between A and B, since the essential content of the proof to be carried out can already be made clear in this special case.

A physical process taking place in system A can, on the one hand, be described by the time-dependent wavefunction ψ A (q A , t) of system A. |ψ A (q A , t )| 2 gives then the probability that at time t = t , the 15 coordinates q A assume the values q A . One will thus assume with probability |ψ A (q A , t )| 2 (up to a constant factor depending on the sensitivity of the measuring instruments) that the measuring devices react when the interaction between A and B is switched on at time t . To the measuring instruments B, C, etc., one applies in this treatment just the laws of classical physics. But on the other hand, one can incorporate also system B into the quantum mechanical formalism by starting with a wavefunction 16 for the total system A + B. From this wavefunction, the probability that system B reacts can then be inferred, and it is to be proved that this probability, up to a constant factor, is again given by |ψ A (q A , t )| 2 .

Before switching on the interaction between A and B, the wavefunction of system A+B is given by the product ψ A (q A , t)•ψ B (q B , t), where ψ B (q B , t) represents the state of system B before the onset of the reaction. We now separate the Hamiltonian function of the total system A + B into the three parts H A , H B and H AB , the first two of which refer to the systems A andB, while H AB denotes the interaction between A and B, which differs from zero only at the point q A = q A and becomes effective only from time t = t . The Schrödinger equation of the system A + B then reads

17 i ∂ ∂t + H A + H B ψ(q A , q B , t) = -H AB ψ(q A , q B , t) . (1)
If one is interested only in the behaviour of the system shortly after time t = t , one can substitute for ψ(q A , q B , t) on the right-hand side of this equation the 'undisturbed' value ψ A (q A , t )ψ B (q B , t ). Further, since the interaction energy H AB differs from zero only at the point q A = q A , the right-hand side can be replaced by H AB ψ A (q A , t )ψ B (q B , t ). From this it transpires that the solution of the Schrödinger equation for times shortly after t = t can be represented in the form

ψ(q A , q B , t) = ψ A (q A , t) ψ B (q B , t) + ψ A (q A , t ) ϕ(q A , q B , t, t ) ,
where the function ϕ(q A , q B , t, t ) no longer depends on the behaviour of system A before time t . The probability that system B has undergone a reaction, that it is therefore no longer in the initial state represented by ψ B (q B , t), is given by the deviation of the wavefunction from ψ A (q A , t) ψ B (q B , t), and thus, according to the rules of quantum mechanics, essentially given by

dq A dq B |ψ A (q A , t ) ϕ(q A , q B , t, t )| 2 = |ψ A (q A , t )| 2 dq A dq B |ϕ(q A , q B , t, t )| 2 .
Thus this 18 probability turns out to be proportional to |ψ A (q A , t )| 2 also ac-

16 [Wellenfunktion/Zahlenfunktion] 17 [ /h in the equation] 18 [Diese/Die]
cording to this second method of calculation, where the remaining constant factor depends only on system B and its interaction with A, but not on the prior history of system A. One could, in the same way, incorporate also system C into the quantum mechanical formalism, and one would arrive at the same end result by very similar calculations. For the prediction of a given physical event, it is thus indeed indifferent at which place one draws the cut between the classical and quantum mechanical treatment.

The content of this proof can be presented in a somewhat more general form as follows. System B is meant to function as the measuring device for the coordinates q A of A. For this it is necessary that, when the interaction between A and B is switched on, a specific value q A corresponds to some specific reaction in B, e.g. the transition of the coordinate q B from the value q B to the value q B . In the language of wave mechanics, this means: when the interaction is switched on, the evolution of the specific solution to the Schrödinger equation with the wavefunction initially differing from zero only in the neighbourhood of q A and q B , is such that 19 the projection of the wave packet onto the space of the q B describes a motion from q B to q B . This is (in this example) the necessary condition for B being a measuring device. Now, before switching on, thus as long as the systems are still independent, the wavefunction is given by the product of the wavefunctions of the subsystems. One can track the temporal evolution of the total wavefunction 20 most easily by constructing it as a superposition of the wave packets just 21 described. The total wavefunction then appears (for a short time after switching on the interaction) as a product of two factors, one of which is given by the wavefunction of the system A to be observed at the time when the interaction is switched on, while the other represents the reaction of the measuring device B. From this follows again the result 22 discussed above. The most essential properties of the quantum mechanical formalism 23 this proof relies on are: first, the fact that any enlargement of the physical system one is describing is represented by an increase in the number of dimensions of the configuration space; second, the possibility of obtaining new solutions through superposition of different solutions to the Schrödinger equation. In addition, in the quantum mechanical representa-tion of the measuring device we have relied on the known 24 relation between geometrical optics and wave optics. However, it follows from this that the classical-theoretical causal connections that are employed in the measuring devices can be reproduced in quantum mechanics only to the degree of precision to which the anschaulich classical variables of the measuring devices can be represented in the wave picture. However, in all practical cases the fundamental indeterminacy brought about in this way by the indeterminacy relations in the formulation of those causal connections is much smaller than the practical uncertainty that must be allowed for in every measuring device, even the best. Devices in which the said fundamental uncertainty plays a larger role would not be capable of being regarded as measuring devices in the ordinary sense, and of being treated according to classical theory. 25 The claim made earlier, that it is indifferent at which location the cut between the parts of the system to be treated quantum mechanically 26 and the classical measuring devices should be drawn, should thus be made more precise in the sense that this cut may indeed be shifted arbitrarily far in the direction of the observer in the region that is otherwise described according to the laws of classical physics; but that this cut cannot be shifted arbitrarily in the direction of the atomic system. Rather, there are physical systems -and all atomic systems belong among these -that the classical concepts are unsuitable to describe, and whose behaviour can therefore be expressed correctly only in the language of wavefunctions.

Wave mechanics is capable of reproducing classical causal connections in the sense just discussed only because its formalism -exactly like that of classical mechanics -itself contains univocal connections [in sich eindeutig zusammenhängt]; in its formalism, the temporal evolution of the wavefunction can be inferred univocally from the initial values of this function with the help of the Schrödinger equation. The statistical element enters quantum mechanical statements only through the 'cut', where the partly un-24 [bekannt/bestimmt] 25 [Marked 'Anm.' in the manuscript, but part of the main text in the transcription:] At this point, attention must be drawn also to a third category of measuring devices, which employ univocal causal connections that, however, cannot be followed in detail within classical theory. As an example, one can take the devices with which one detects the presence of neutrons in a specific position by generating artificial radioactivity. The physical processes that take place between the capture of the neutron and the creation and emission of the electron cannot be described in the framework of classical theory. Nevertheless, we have here a measuring device that allows for univocal inferences and can for instance play the role of system B in the proof we have discussed.

26 [den quantenmechanisch zu behandelnden/dem quanten-mechanisch zu behandelnden]

controllable 27 disturbance necessarily tied to every observation can be held responsible for the appearance of statistical connections. §2. After these preliminaries we return to the question posed at the beginning, whether the statistical predictions of quantum mechanics are capable of a deterministic completion. We shall thus investigate the assumption that the physical systems about which quantum mechanics makes statistical statements are bearers of hitherto unknown physical properties that determine univocally the behaviour hitherto known only statistically. For example, let the nucleus of the radium atom possess, apart from the properties that are fixed by the knowledge of its stationary state, still other hitherto unknown properties whose knowledge makes possible an exact prediction of when the atomic nucleus will emit an α-particle. And we assume specifically: this statement should hold independently of the means of observation that are used to detect the α-particle. 28

One might at first be tempted to drop this last special assumption, and consider that the radioactive emission may also depend on the properties of the means of observation. But this gain in generality is only apparent. For instance, one could count as part of the system to be observed the counter device and the photographic plate that bf instead of: which registers the events [ihre Anschläge]. Then the question of the blackening of the plate replaces the question of the emission of the α-particle. And here, one is surely forced to assume that whether or not the blackening of the plate occurs is entirely independent of how the observer later takes a look at the plate. For our description of nature is based in the end on the assumption that one can speak of objective events in space and time. It would be altogether impossible to speak of the correctness of the predictions of any theory if one did not assume that the occurrence of a particular event is an objective fact that does not depend on our observation of this event. Classical physics defines the domain 29 within which we can objectify our perceptions unproblematically. -If one considers the blackening of a photographic plate to be an objective fact in this way, then there is actually no further reason not to consider also the emission of an α-particle to be an objective fact; except, that is, if at some place between the macroscopic and the microscopic events one assumes there is a discontinuity for which not the slightest indications Accordingly, we must discuss the special assumption that a physical system, for which quantum mechanics allows one to predict only 30 the probability for the occurrence of a specific event, possesses still other hitherto unknown properties, knowledge of which would make precise statements possible about the occurrence of this event -independently of how the occurrence of the event is observed. It shall now be shown that this assumption runs into contradiction with the statements of quantum mechanics. And in fact not just with the statistical statements of quantum mechanics; this would not be surprising, for the observation of the hitherto 31 unknown properties could make it possible to select specific systems from a quantum mechanical ensemble, and one cannot expect that the same statistics hold for individual systems selected according to specific points of view as for the ensemble. Rather, the said assumption also comes into contradiction with specific statements of quantum mechanics.

The reason for this contradiction lies in the already discussed fact that the formalism of quantum mechanics itself establishes a univocal connection between the quantities that it relates to each other, and that the quantum mechanical statements acquire their statistical character only because at the location separating the observer with his devices from the system to be observed, a fundamentally 32 uncontrollable disturbance of the system by the means of observation prevents us from following the causal connections. Let us thus place 33 the cut in the above discussed scenario between, say, the atomic system A and the device B. Then we would have to ascribe to system A hitherto unknown properties that complement with specific statements the statistical statements that are to be made about A according to quantum mechanics. But now there is no reason within the framework of quantum mechanics for not also placing the cut between systems B and C rather than between A and B. Then the systems A and B are univocally connected in the formalism of quantum mechanics, and every statement about A not already contained in this univocal connection can run into contradiction with what this connection implies.

In detail, the contradiction arises in the following way: on the one hand, it is meant to follow from the knowledge of the hitherto hidden physical properties of system A, that A in its interaction with34 any physical system, thus in particular with B, reacts as if the coordinate q A had the value q A . But on the other hand B can be so chosen that according to quantum mechanics its reaction to A proceeds differently. For instance (as opposed to the above discussed example) B can measure a property of A complementary to q A . Then the possibility of a reaction of system B, as determined by quantum mechanics, rests precisely on the freedom that still obtains in the value of q A ; thus B's reaction to A is different to what would follow from the coordinate q A having the value q A . More generally:35 the quantum mechanical formalism that is applied to the total system A + B contains no freedom whatsoever, and leaves no place for additional assumptions regarding the effect of A on B; just as little as the mathematical framework of classical mechanics would. There would be room for complementing the quantum mechanical statements only at the location of the cut; but this location cannot be determined physically, indeed, it is precisely the arbitrariness in the choice of the location of the cut that is crucial for the application of quantum mechanics. Any hitherto unknown physical properties, which would then be necessarily tied to their specific physical system, are thus fundamentally unsuitable for complementing the quantum mechanical statements.

One can easily illustrate this state of affairs with the above mentioned example of the radioactive atomic nucleus. The α-particles leave the atomic nucleus with an energy that is fixed up to quantities of the order /T , if one takes T to be the average decay time of the atomic nucleus. One can let this α-radiation impinge on a diffraction grating of very high resolution, and according to the laws of quantum mechanics, one can be certain that (if the decay time is not too short) the α-radiation is reflected by the grating only in one of the extraordinarily sharply determined directions that according to the laws of interference correspond to the de Broglie wavelength of the αrays. Now, if there were hitherto unknown physical properties of the atomic nucleus allowing one to predict in which direction and at what moment the α-particle is emitted, then one could calculate what part of the diffraction grating is hit by the α-particle, and when it is hit. The direction of reflection of the α-particle could then (up to negligibly small corrections) only depend on the properties, at the time of the impact, of that part of the grating that is hit, because whatever the interaction, the α-particle ought to be found at the predicted36 position in question -that was the assumption made above -, thus it cannot interact with the distant parts of the grating. On the other hand, according to quantum mechanics, it is precisely the distant parts of the grating that are crucial in determining the direction of reflection. Thus a contradiction arises. If one replaces the radioactive atomic nucleus in the above discussed example with a sodium atom emitting light belonging to the D-line, one recognises immediately that this contradiction does not arise only in thought experiments that are hard to verify. A definite prediction of the position at which the light quantum hits the diffraction grating would be in contradiction with the occurrence of a sharp diffraction phenomenon.

However, the strength of this example37 rests crucially on the assumption that the localisation of an α-particle or light quantum applies to every kind of interaction, that one can thus speak of the location of the α-particle with the same right as of the location of a macroscopic object. If one does not accept this, then one should apply the same argument to a system for which there can be no doubt about the applicability of classical anschaulich concepts, thus one should include a part of the macroscopic measuring devices38 in the system.

With this argument one can also counter the objection that inferences to the statistical character of atomic physics are not conclusive, because the concepts there used of corpuscle and wave, light quantum and field are in fact unsuitable for describing the physical conditions, and therefore will have to be replaced later by other conceptions. This objection overlooks that in the end the application of the quantum mechanical formalism does not require at all the use of these concepts, thus also that nothing is gained by eliminating these concepts. For quantum mechanics can always be treated such that part of the measuring devices can be counted as part of the system to be observed, and represented by wavefunctions. Then, the results of the quantum mechanical calculation include also anschaulich statements about such things as pointer positions, blackening of plates and the like, in the case of which one cannot doubt one is already using the correct concepts for describing the physical conditions. It is a crucial feature of quantum mechanics that its formalism allows us to link organically the physical domains that are in principle beyond the reach of our Anschauung with the macroscopic anschaulich domains, in such a way that the statements of the formalism can be unambiguously expressed through anschaulich concepts.

Of course, quantum mechanics expressly assumes, just as in the argument carried out here, that it is possible after all to objectify our perceptions39 at some point, i.e. to speak of things and processes. Classical physics proves that this is possible in a wide domain, and the whole of natural science rests upon this possibility. §3. The question posed at the beginning, whether a deterministic completion of quantum mechanics is possible, is thus to be answered in the negative. If one wants to express the reasons put forward for this in a simple formula, then according to G. Hermann,40 one can say that a deterministic completion of quantum mechanics is impossible for the reason that quantum mechanics already allows the complete specification of the causes for the occurrence of a definite measurement result. From this formulation the question immediately arises, which feature of nature -obviously overlooked by classical physics -is actually responsible for the fact that the formalism of quantum mechanics, itself containing univocal connections [in sich eindeutig zusammenhängend], is not sufficient for calculating in advance all measurement results, that therefore statistical connections appear at the location of the cut. One recognises this feature best from a simple example: in the classical theory one could say of a particle that it moves with a definite velocity at a definite position. If one wants to translate this statement (with the inherent unavoidable unsharpness) into quantum theory, then one will say that a wave packet in configuration space is moving with a definite velocity at a definite position. Thereby, however, the state is not yet uniquely fixed. Indeed, to fix it uniquely one must supply still further data about the size and form of the wave packet, for which there are altogether no analogies41 in classical theory. Quantum mechanics has thus revealed to us here a new property of nature that was unknown to classical physics.

Since knowledge of the quantum mechanical wavefunction can be gained from a suitable observation of the system and is enough to determine for later times which observations on the system will have results that can be predicted exactly,42 one can speak of an 'observational context' specified43 by knowledge of the wavefunction. The example just discussed shows that the same anschaulich process can belong to different observational contexts -as opposed to classical physics, in which there is only a single observational context. The experimental results accumulated in quantum mechanics have further shown that the observation of a system in general leads discontinuously from one observational context to another. The causal flow can be followed only within a given observational context; in the discontinuous transition from one observational context to another (in fact a 'complementary' one in the Bohrian sense), only statistical predictions are possible. The possibility of different complementary observational contexts, unknown to the classical theory, is thus responsible for the occurrence of statistical laws.

After one manages to see that a deterministic completion of quantum mechanics would lead to contradictions, one can pose the further question whether a later modification of quantum mechanics might not open up the possibility for such a deterministic completion. In attempting to answer this further question, one can naturally only call upon the experimental results. One can verify that the conclusions of §2 essentially make use only of the results of quantum mechanics that have been repeatedly confirmed experimentally; the most important example is given there by the experimentally established fact that the radiation emitted by an atom can give rise to interference phenomena, even though the total energy of the radiation is eventually absorbed at a definite point (as a 'light quantum'). Furthermore, one can point to the fact that the statistical44 character of quantum mechanics is very tightly bound to the formal circumstance that its mathematical framework of wavefunctions operates in multi-dimensional configuration space, not in ordinary space, and that precisely this feature of quantum mechanics has been exactly confirmed through the correct reproduction of the more complicated atomic spectra. Yet of course it can never be decided when a sum of experimental results licenses conclusive inferences to a law.

Here for the moment one will have to be content with the observation that neither the experimental results nor considerations of principle give one grounds45 to believe that the future description of nature will let itself be fitted into the narrow classical framework of an anschaulich and causal

  Heisenberg: Die physikalischen Prinzipien der Quantentheorie. Leipzig 1930; J. v. Neumann: Mathematische Grundlagen der Quantenmechanik. Berlin 1932; W. Pauli: Die allgemeinen Prinzipien der Wellenmechanik. Handbuch der Physik, Vol. 24, 1st Part. Berlin 1933. 10 [Wellenfunktionen/Wellenfunktion] 11 [seiner/dieser]a line on a photographic plate, is taken to be a classical-anschaulich process? At what place should one draw the cut between the description by wavefunctions and the classical-anschaulich description?

  12 [eben/oben] 13 [sein/mein] 14 [aus/von] 15 [Missing 'die' in the transcription.]

  27 [teilweise unkontrollierbare/teilweise nur kontrollierbare] 28 [des α-Teilchens/der α-Teilchen] 29 [Bereich/Beweis] are present either in experience or in the quantum mechanical formalism.

  30 [nur/und] 31 [bisher/beiden] 32 [prinzipiell/prinzipielle] 33 [Legen wir also/Leben wir also]

  , section 7, 'Der Zirkel in Neumanns Beweis'. Available at http://www.phys.uu.nl/igg/seevinck/trans.pdf . Soler, L. (2009). The convergence of transcendental philosophy and quantum physics: Grete Henry-Hermann's 1935 pioneering proposal. In M. Bitbol, P. Kerszberg, and J. Petitot (eds.), Constituting objectivity: Transcendental perspectives on modern physics. The Western Ontario series in philosophy of science,Vol. 74. Berlin: Springer,

Heisenberg to Einstein, 10 June 1927 (Albert Einstein Archive 12-174.00, in German).

Hermann's paper is one of the earliest and best philosophical treatments of the new quantum mechanics. Some of her views have been discussed byJammer (1974, pp. 207- 208), but very little of her work has been translated into English. See Seevinck (no date) for an online translation of Hermann's section 7, which criticises von Neumann's no-hidden-variables proof, and[START_REF] Hermann | Les Fondements philosophiques de la mécanique quantique. Introduction and commentary[END_REF] for a complete translation into French with an extensive introduction and commentary by L. Soler (see alsoSoler 2009).[START_REF] Hermann | The foundations of quantum mechanics in the philosophy of nature. Translated and with brief introduction by D[END_REF] is an English translation by D. Lumma of Hermann's shorter paper for Die Naturwissenschaften(Hermann 1935b).

[so, dass/so daß]

[Gesamtwellenfunktion/Gesamtwellenfunktionen]

[eben/etwa]

[Ergebnis/System]

[des quantenmechanischen Formalismus/der Quantenmechanik den Formalismus]

[Spurious 'B' in the transcription.]

['Allgemeiner:' missing in the transcription.]

[vorhergesagten/vorhergegangenen]

[dieses Beispiels/dieser Beispiele]

[der makroskopischen Messapparate/des makroskopischen Meßapparates]

[Wahrnehmungen/Wechselwirkungen]

G. Hermann: Die [natur]philosophischen Grundlagen der Quantenmechanik. Berlin 1935. [Title amended already in the transcription.]

[keine Analogien/kein Analogon]

[exakt/nicht]

[charakterisierten/charakteristischen]

[statistische/natürliche]

[Anlass bieten/Anlaß bilden]

description of objective processes in space and time.