M Clusel

J.-Y Fortin

Grassmann techniques applied to classical spin systems

Keywords: Grassmann algebra, spin systems, critical phenomena PACS: 02.30.Ik, 05.50.+q, 05.70.Fh

We review problems involving the use of Grassmann techniques in the field of classical spin systems in two dimensions. These techniques are useful to perform exact correspondences between classical spin Hamiltonians and field-theory fermionic actions. This contributes to a better understanding of critical behavior of these models in term of non-quadratic effective actions which can been seen as an extension of the free fermion Ising model. Within this method, identification of bare masses allows for an accurate estimation of critical points or lines and which is supported by Monte-Carlo results and diagrammatic techniques.

Introduction

Classical an quantum spin models such as Ising model play an important role in the field of statistical physics as they allow for an accurate understanding of critical phenomena in general. Many techniques [ 1,2] were developed in order to deal with the difficulty of estimating the partition function and other thermodynamical properties in the critical region in dimension more than one. An exact mathematical description of the two-dimensional (2D) Ising model relies on the Jordan-Wigner transformation [ 3] which maps the product of Boltzmann weights onto a fermionic action of free fermions with a mass vanishing at the second order critical temperature given in dimensionless units T c = 2/ ln(1+ √ 2) ≃ 2. 2691851. Also a method based on the correspondence between the Ising model and dimer problems [ 4] uses the notion of Pfaffians, which are directly connected to integrals over Grassmannian objects.

Both fermions and Grassmann variables are therefore closely tied to the Ising model.

A direct introduction of Grassmann variables as an alternative tool to solve the Ising model was done long ago in the 80's by Bugrij [START_REF]Bugrij in Electron-Electron Correlation Effects in Low-Dimentional Conductors and Superconductors[END_REF] and Plechko [START_REF] Plechko | [END_REF] (see also a later discussion by Nojima [ 7]). It is based on a simple integral representation of the individual Boltzmann weights and which has the property to decouple the spins.

The price to pay is a non-commutativity of terms arising from this representation.

In order to deal with this particular representation, Bugrij used two families of Grassmann variables which commute with each other, then identified the resulting functional integral of the partition function with a determinant. From another point of view, Plechko introduced symmetries which order the non-commuting quantities so that the sum over the spins can be performed exactly. In this paper we review the process of how to generalize Plechko's method for Blume-Capel model [ 8], which is the simplest model beyond Ising, to spin-S Ising models and how to construct an exact fermionic action for each case. This would provide a natural extension of the exact fermionic quadratic action found for the spin S = 1/2 Ising model. In particular, we will build on previous work on the Blume-Capel (BC) case S = 1 [ 8] where a line of second-order critical points is terminated by a tricritical point. This is the next case beyond the Ising model and which possesses a rich critical behavior.

This model was used to qualitatively explain the phase transition in a mixture of He 3 -He 4 adsorbed on a 2D surface [ 9]. Below a concentration of 67% in He 3 , the mixture undergoes a λ transition and the two components separate through a first order phase transition with only He 4 superfluid. On a 2D lattice, He atoms are represented by a spin-like variable, according to the following rule: an He 3 atom is associated to the value 0, whereas a He 4 is represented by a classical Ising spin taking the values ±1. Within this framework, all the lattice sites are occupied either by an He 3 or He 4 atom. In addition to nearest-neighbor interactions, the energy includes a term ∆ 0 mn S 2 mn , with S 2 mn = 0, 1, to take into account a possible change in vacancies number. ∆ 0 can be viewed as a chemical potential for vacancies, or as a parameter of crystal field in a magnetic interpretation of atomic physics. It would be in particular interesting to have a fermionic description of the BC model in order to obtain more information about the kind of interaction fermions living on the 2D lattice have compare to the Ising free fermion case.

So one of the main question is how to obtain a generic fermionic action for a spin-S model and what does this method teaches us for the BC model in particular.

We explain in the next section the main ideas of this method.

Description of the fermionization for general classical spin-S models

Let us consider the following Hamiltonian on a 2D lattice of size

L × L H = - L m=1 L n=1 J S mn S m+1n + S mn S mn+1 + ∆ 0 L m=1 L n=1 S 2 mn , (1)
where J is the Ising coupling constant and ∆ 0 the splitting crystal field or represents a chemical potential in the BC case. In particular for ∆ 0 large and positive, it favors small spin values. This crystal field can be replaced by any potential V (S 2 mn ) depending on the square of the local spin. Spins S mn take 2S + 1 values with S mn = -S, -S + 1, • • • , S. The partition function is the sum over all possible spin configurations Z = Tr exp(-βH). Z contains products of the Boltzmann weights exp(KS mn S m+1n ) (where S mn and S m+1n are neighboring spins and K = J/k B T ) which take q + 1 = S(S + 1) + 1 different values if S is an integer, and q + 1 = (S + 1/2)(S + 3/2) values if S is half-integer. Since there are q + 1 possible values for each Boltzmann weight, we can project each of them onto a polynomial function of degree q in the variable S mn S m+1n :

exp(KS mn S m+1n ) = q k=0 u k (S mn S m+1n ) k = u 0 q α=1 (1 + x α S mn S m+1n ), (2)
where the q + 1 constants u k are determined by solving the linear system of q + 1 equations satisfied by the above relation. To see on specific examples how it works, let us consider first the Ising case, S = 1/2. Since S is half integer, we have q = 1.

Therefore exp(KS mn S m+1n ) = ch(K/4) + 4 sh(K/4)S mn S m+1n , u 0 = ch(K/4), u 1 = 4 sh(K/4). (3)

In the Blume-Capel model, since S is integer, we have q = 2 and it is straightforward to show that exp(KS mn S m+1n ) = 1 + sh(K)S mn S m+1n + (ch(K) -1)(S mn S m+1n ) 2 ,

u 0 = 1, u 1 = sh(K), u 2 = ch(K) -1. ( 4
)
For S integer the first coefficient u 0 is always unity, and from equation (2) we can write

u k = u 0 α 1 <α 2 <•••<α k x α 1 x α 2 • • • x α k , 1 ≤ k ≤ q. ( 5
)
We will set for convenience in the following u k≥q+1 = 0 since the polynoms are all finite. Our purpose is to transform the partition function Z which is a sum over spin variables into a multiple integral over Grassmann variables. For this let us introduce q pairs of Grassmann variables [ 8,[START_REF] Plechko | [END_REF]10] (a α mn , āα mn ) on each site for the horizontal direction and q additional pairs (b α mn , bα mn ) for the vertical direction. Here α takes the values 1, . . . q. There are therefore 4q Grassmann variables at each site of the lattice.

In particular the Ising model is represented by two pairs of Grassmann variables per site which can afterward be reduced to only one pair [START_REF] Plechko | [END_REF]. For each couple of terms (1 + x α S mn S m+1n )(1 + x α S mn S mn+1 ) [START_REF] Plechko | [END_REF] appearing in the partition function, we introduce the following integral representation 1 + x α S mn S m+1n = dā α mn da α mn e a α mn āα mn (1 + a α mn S mn )(1 + x α āα mn S m+1n ),

1 + x α S mn S mn+1 = d bα mn db α mn e b α mn bα mn (1 + b α mn S mn )(1 + x α bα mn S mn+1 ). ( 7
)
From the last expression, we introduce the link factors A α mn = 1 + a α mn S mn , Āα m+1n = 1 + x α āα mn S m+1n , B α mn = 1 + b α mn S mn , and Bα mn+1 = 1 + x α bα mn S mn+1 , so that the partition function can be written as

Z = u 2L
can be moved freely with the other terms since they correspond to commutative scalars after integration. In particular, it is convenient to rearrange the products over α in order to put together the link factors of different α with the same site indices (m, n) using the mirror ordering symmetry introduced in Plechko's method [START_REF] Plechko | [END_REF] in the context of the 2D Ising model, and which is still relevant in the spin-S case:

q α=1 (A α mn Āα m+1n ) = (A 1 mn Ā1 m+1n ) . . . (A q mn Āq m+1n ), = (A 1 mn (A 2 mn . . . (A q-1 mn (A q mn Āq m+1n ) Āq-1 m+1n ).. Ā1 m+1n ), =   -→ q α=1 A α mn     ← - q α=1 Āα m+1n   ,
where the arrows indicate that the product is ordered, i.e. increasing label α in the first product from left to the right and in the second one from right to the left. For convenience, we will use the notation

O mn = -→ α A α mn , Ōm+1n = ← - α Āα m+1n , (9)
for objects on the horizontal links and

P mn = -→ α B α mn , Pmn+1 = ← - α Bα mn+1 (10)
for the ones on vertical links. Then the partition function can be rewritten as

Z = u 2L
At this stage of the algebra, we use the mirror and associative symmetries which were used for solving the Ising model [START_REF] Plechko | [END_REF]10] and which are still valid here to rearrange the operators O and P. In principle boundary terms should be treated separately in order to obtain the exact finite size partition function depending on boundary conditions [START_REF] Plechko | [END_REF] but they are not relevant in the thermodynamical limit L → ∞ we are interested in here. Here we consider instead the simple case of free boundary conditions, and we obtain the exact expression after rearrangement of the O and P operators:

Z = u 2L 2 0 Tr {S,a,b}   -→ L n=1 -→ L m=1 e ∆S 2 mn Ōmn Pmn O mn ←- L m=1 P mn   .
Now, from this expression, the spins can individually be summed up from S Ln to S 1n for any given n. We will need to introduce the following weights W mn which include all the dependence on the individual spins S mn

W mn = Smn=±1 e ∆S 2 mn Ōmn Pmn O mn , P mn , ≡ Smn=±1 e ∆S 2 mn -→ 4q α=1 1 + c α mn S mn , (12)
where we have defined the following 4q sets of Grassmann variables c α mn in the following order:

c 1 mn = x q āq m-1n , c 2 mn = x q-1 āq-1 m-1n , • • • , c q mn = x 1 ā1 m-1n , c q+1 mn = x q bq mn-1 , c q+2 mn = x q-1 bq-1 mn-1 , • • • , c 2q mn = x 1 b1 mn-1 , c 2q+1 mn = a 1 mn , c 2q+2 mn = a 2 mn , • • • , c 3q mn = a q mn , c 3q+1 mn = b 1 mn , c 3q+2 mn = b 2 mn , • • • , c 4q mn = b q mn . (13)
The sum over S mn = ±1 in equation ( 12) can be performed by noticing that only products involving an even number of S mn give a non-zero contribution. We also define the scalars (we remind that ∆ = -β∆ 0 )

α k = S Smn=-S S 2k mn exp(∆S 2 mn ), (14)
and the ordered products

q (k) mn = α 1 <α 2 <•••<α k c α 1 mn c α 2 mn • • • c α k mn , q (0) mn ≡ 1, (15)
with q (4q)

mn = c 1 mn • • • c 4q
mn the term of highest degree in Grassmann variables. Using these quantities, it is easy to show that the partial Boltzmann weights [START_REF] Berezin | The Method of second quantization[END_REF] are given by

W mn = 2q k=0 α k q (2k) mn . (16)
Then the fermionic representation of the partition function can be expressed as a multiple integral over Grassmannian variables only

Z = u 2L 2 0 Tr {a,b,ā, b} mn W mn . (17)
For small values of S, the weights W mn can be exponentiated so that a fermionic action can be defined. Indeed, since the first term of W mn is the pure scalar α 0 and the others products of pure commutating Grassmannian objects, it is tempting to exponentiate the sum ( 16) to obtain directly a fermionic action. This comes from the simple observation that for any Grassmann variable a, we have 1 + a = e a . Of course, the exponentiation of the sum ( 16) is more complicate. For example, for commuting objets a and b such as the q k mn s, we have 1 + a + b = exp(1 + a + bab). In this case the order of the polynomial object inside the exponential is bigger than in the original sum since the extra counter-term ab is necessary for the identity to be exact. These weights are moreover connected by nearest-neighbor interactions hidden in the variables c α mn . In the case of the Ising model, where the exponentiation can be done quite easily, the argument of the exponential is purely quadratic in the c α mn 's and therefore the partition function can be integrated out with the use of a determinant or a Bogoliubov transformation in the Fourier space. Moreover, the 4q = 4 Grassmann variables in this case can be reduced to 2 by partial integration of non relevant variables. In the BC model, the argument is a polynomial of degree 8 in Grassmann variables since there are 8 independent variables (4q = 8). In general we expect naturally the argument to be at most a polynomial of degree 4q in these variables, which can be reduced or not by partial integrations. Except for the case q = 1 however the partition function can not be expressed as a determinant, so that a full exact solution of the partition function can not be found this way. If the action is quadratic, the use of the following Gaussian integral [START_REF] Berezin | The Method of second quantization[END_REF], defined on Grassmann set of variables {a i , āi } i=1,..,N , and for a square matrix

A N i=1 dā i da i exp N i,j=1 a i A ij āj = det A , (18)
allows us to express the partition function as a determinant. Quadratic fermionic form in the exponential ( 18) is typically called action for a free-field theory. When the action is non-quadratic, the integral is not Gaussian and can not be expressed as a determinant, which yields in principle to a non integrable theory. However, physical information such as bare masses (see last section) can be extracted from these non-quadratic actions which represent generic theories of interacting fermions.

Fermionic action of the Blume-Capel model

In this section, we consider the case S = 1 (Blume-Capel model) which is the simplest example of a classical spin beyond the Ising model. It possesses in the phase diagram (T, ∆ 0 ) a second-order critical line separating a ordered phase from a disordered one and terminated by a tricritical point (see figure 1 below). From the previous section equation ( 17) allows us to write an action after exponentiation of the Grassmann variables which can be done exactly after some tedious algebra. The 4 pairs of variables per site can however be reduced to 2 pairs by partial integration.

Another simpler way of obtaining this BC fermionic action is possible [ 11] using the Z 2 symmetry of the spin variables S mn . Indeed the partition function is invariant if we perform the gauge transformation S mn → σ mn S mn with σ mn = ±1. In this case it is possible to simplify the process of the previous method and write an action containing only 2 pairs of variables per site instead of 4:

Z = (2e ∆ cosh 2 K) L 2 L m=1 L n=1 dā mn da mn d bmn db mn exp L m=1 L n=1 (19)
a mn āmn + b mn bmn + a mn b mn + t(ā m-1n + bmn-1 )(a mn + b mn ) + t 2 ām-1n bmn-1 + g 0 a mn āmn b mn bmn exp (-γa m-1n ām-1nγb mn-1 bmn-1t 2 ām-1n bmn-1 ) ,

where we have introduced the following constants:

g 0 = e -∆ 2 cosh 2 K , γ = 1 - 1 cosh K = 1 - √ 1 -t 2 , t = tanh K . ( 20
)
The fermionic integral (19) is the exact expression even for a finite lattice, provided we assume free boundary conditions for both spins and fermions. The other possible form for the partition function with periodic boundary conditions in both direction can be written in a similar way as the Ising model on a torus [START_REF] Plechko | [END_REF][START_REF] Liaw | [END_REF]14]. The partition function would be the sum of 4 fermionic integrals with periodic-antiperiodic boundary conditions for the fermions. In the expression (19), we can recognize the sum of the Ising action, which here appears as the Gaussian part of the total action [START_REF] Plechko | [END_REF]10]:

S Ising = L m,n=1
a mn āmn + b mn bmn + a mn b mn + t(ā m-1n + bmn-1 )(a mn + b mn ) + t 2 ām-1n bmn-1 , and a non-quadratic interaction part, which is a polynomial of degree 8 in Grassmann variables (which can be seen if we expand the exponential inside the action):

S int = g 0 L m,n=1
a mn āmn b mn bmn expγa m-1n ām-1nγb mn-1 bmn-1t 2 ām-1n bmn-1 . (21) This allows us to rewrite the partition function as a fermionic field-theory in a compact form

Z = (2e ∆ cosh 2 K) L 2 DāDaD bDb exp(S Ising + S int ) . (22)
The BC model differs from the Ising model by the interaction term in the action (21) which is not quadratic. Therefore the BC model is not solvable in the sense of free fermions as a determinant of some matrix, unlike the 2D Ising model.

Mixed representation of the BC model

The coupling of Grassmann variables in equation ( 21) prevents us to integrate further and reduce the number of variables per site unlike the Ising model where the minimal action contains one pair only [ 18,7]. The minimal action of the Ising model admits an interpretation in term of Dirac representation of free fermions which become massless at the critical point. In a previous work we were able to reduce the number of Grassmann variables by partially introducing hard core bosons in the previous action, since terms such as η mn = a mn āmn or τ mn = b mn bmn may have an interpretation of local densities or occupation numbers. Variables η mn and τ mn are commuting and nilpotent, η 2 mn = τ 2 mn = 0. We can replace the quantities depending on a mn āmn and b mn bmn , especially in the interaction part, by their respective nilpotent variables, using, for this task, a general definition of Dirac distribution for any polynomial function f of a mn āmn or b mn bmn [ 11]: f (a mn āmn ) = dη mn dη mn f (η mn ) exp [η mn (η mn + a mn āmn )] , f (b mn bmn ) = dτ mn dτ mn f (τ mn ) exp τmn (τ mn + b mn bmn ) .

A natural definition [ 15] of the integrals involving commuting nilpotent variables is to impose the following rules (and similar for ηmn , τmn ):

dη mn (1, η mn ) = (0, 1) , dτ mn (1, τ mn ) = (0, 1) .

This change of variables allows us now to integrate over the a mn 's and b mn 's in the new action. One advantage is that after this operation there are only two fermionic variables per site, although two additional pairs of bosonic variables have been introduced. In fact we can integrate over one pair of bosonic variables [ 11], for example ηmn , τmn , using the help of integration rules and Dirac function given by [START_REF] Mattuck | A Guide to Feynman Diagrams in the Many-body Problem[END_REF]. At the end, it remains a mixed action made of one pair per site of fermionic and bosonic variables respectively, with an interaction between fermions and bosons. A convenient replacement of the variables āmn by c mn and bmn by -c mn in the final integral leads us to isolate the minimal local action for the pure Ising model [START_REF] Plechko | Anticommuting integrals and fermionic field theories for twodimensional Ising models[END_REF][START_REF] Plechko | [END_REF] with one pair of Grassmann variables per site:

S Ising = c mn cmn + t(c mn + cmn )(c m-1n -cmn-1 ) -t 2 c m-1n cmn-1 , (25)
and the interaction part

S int = g 0 m,n η mn τ mn (1 -γη m-1n )(1 -γτ mn-1 ) + t 2 c m-1n cmn-1 , (26)
with the quantities qmn = c mn cmn + tc mn (c m-1ncmn-1 ) = c mn [c mn + t(c m-1ncmn-1 )] ,

q mn = c mn cmn + tc mn (c m-1n -cmn-1 ) = [c mn -t(c m-1n -cmn-1 )]c mn . ( 27
)
The Ising part is the same action that results from the integration over a mn , b mn from the original Ising case. The introduction of nilpotent variables was necessary to achieve this partial extraction of the Ising contribution. The physical interpretation of the previous mixed representation is that it can be possible to describe the BC model with fermionic variables for the states S = ±1 and bosonic ones for states S = 0. In the limit ∆ 0 → -∞, the system is completely described in terms of fermions (Ising sector), while when ∆ 0 is increasing fermions and bosons begin to interact. Beyond a critical value of ∆ 0 , fermions form bosonic pairs and in the limit ∆ 0 → +∞, all fermions condense into bosons, leading to a purely bosonic system.

This view should be supported by further analysis.

Corrections to the effective action in the continuum limit

The integration of the previous action (26) over variables (η mn , τ mn ) can be performed perturbatively, as part of an expansion in the low momentum limit. We will define formally the derivatives of Grassmann variables [ 18],

∂ x c mn = c mn -c m-1n
and ∂ y c mn = c mnc mn-1 in the limit of large L. In this limit and in the Fourier space, the high order derivatives account in the action for a small contribution in momenta k = 2π(m, n)/L, with m, n ≪ L positive integers. We would like to obtain in this limit the non trivial part of the non-quadratic interaction in term of variables c mn , cmn only. The procedure is described in reference [ 11] and based partially on substitution rules such as η mn τ mn → c mn cmn , η mn → qmn , τ mn → q mn .

There are unfortunately more complicate terms in the resulting action than by using the substitution rules alone, such as

g 2 0 γ 2 c mn cmn c m+1n cm+1n c mn+1 cmn+1 , (29)
but they can be discarded in the approximation scheme above in the sense they correspond to corrective terms higher than quartic polynomials or quantities of the order of O(g 0 ) where g 0 is the natural parameter of the expansion. It is exponentially small in the region where ∆ 0 is large and negative (Ising behavior). At the lowest order we found that the effective action [START_REF] Butera | [END_REF]26) can be approximated by the following expansion with respect with g 0

S effective = S Ising + g 0 m,n c mn cmn (1 -γ qm-1n )(1 -γq mn-1 ) + t 2 c m-1n cmn-1 +g 2 0 γ 2 m,n c mn cmn c m+1n cm+1n c mn+1 cmn+1 + . . . . ( 30
)
From the previous result, it appears to be suitable to express the quadratic and quartic parts in the Fourier space (in the large but finite L limit), where we define the following transformations

c(r) = 1 L k c k exp(ik.r) , c(r) = 1 L k ck exp(-ik.r) . (31)
The Ising part of the action can be written as

S Ising = k∈S [m BC + it(t + 1)(k x -k y )](c k ck -c -k c-k ) + 2itk x c k c -k + 2itk y ck c-k ,( 32
)
with m BC = 1 -2tt 2 + g 0 and the quartic term can be express as

S int = g 0 1 L 2 k ′ ,k ′′ ,q V k ′′ ,k ′′ -q c k ′ c k ′′ ck ′ +q ck ′′ -q , (33)
with the potential

V k,k ′ = -αk x k ′ y + α ′ (k x k ′ x + k y k ′ y ), α = t(t + 2γ) , α ′ = γ(1 -t) . ( 34
)
We notice that the bare mass of the theory is given by

m BC = 1 -2t -t 2 + g 0 = m Ising + g 0 , (35)
where m Ising = 1 -2tt 2 is the Ising mass which vanishes at the critical value tanh(K c ) = √ 2-1 corresponding to the second order transition point T c = 2/ ln(1+ √

2) ≃ 2. 2691851 in units of J/k B . In the BC model, the critical temperature is shifted by the parameter g 0 which depends on the temperature and ∆ 0 . The location of the second order critical line goes from the previous Ising critical value

T c = 2/ log(1 + √ 2) when ∆ 0 → -∞ to the zero temperature point (T c = 0, ∆ 0 = 2)
continuously where the transition can be proved to be first order by a simple energetic argument. In figure (1), we have reported the critical line given by m BC = 0 and the different numerical results found in literature [ 11]. In general the agreement is good, which validates the fermionic theory giving a bare mass vanishing at locations close to critical point values found by numerical methods. The presence of a tricritical point is induced by the interaction term (34) which renders the second order line instable. To see why, let us consider the infrared limit on the critical line.

The spectrum is given by the lowest terms of an expansion of the effective action with respect with kinetic terms, an the contribution to the partition function, in the Fourier space, is the product of partial integrals Z k such as Z = k Z k , up to the second order in the momentum k. For the Ising model and for small momenta, the factors Z k are exactly of the form (m Ising + Ak 2 ), with A a constant equal to t(1t 2 ) [ 18]. The coefficient in front of the term k 2 in factors Z k can by described physically as a stiffness coefficient. For the Ising model, the stiffness is always strictly positive even at the critical point. In the BC case, however, we have a line of critical points as ∆ 0 varies from negative to positive values up to ∆ 0 = 2. The effect of the interaction potential (34) is to modify the expression of the stiffness, which now is no more constant but depends on the angle of the vector k and also the temperature and ∆ 0 (see reference [ 11] for explicit details). The result is that in the BC case the effective stiffness coefficient vanishes at some point on the critical line, at a value close to ∆ 0 = 2, which indicates eventually the presence of a tricritical point. It can be shown that the partition function can be indeed written as a product over the

13 MC-JYF Fourier modes Z = k∈S Z k with Z k = m 2 BC + k 2 [A + B sin 2θ k ], (36)
θ k being the angle of the vector k, and A and B depending on temperature and ∆ 0 .

As long as |A| is larger than |B| on the critical line, the transition is second order. A singular point can be reached if A 2 = B 2 , in such case Z k are not all strictly positive if m BC = 0. Beyond this point the effective action (30) is not sufficient to describe the critical properties of the model. If we compare the fermionic description of the BC model to a bosonic Ginsburg-Landau Φ 6 theory describing first order transitions, the presence of a tricritical point would be equivalent to the fact that both coefficients of Φ 2 and Φ 4 terms vanish.

Critical behavior of the BC model: diagrammatic expansion

In this section, we further analyze the influence of the interaction potential V k,k ′ on the renormalized mass, in particular the shift of the critical temperature which was in reference [ 11] assumed to be given by the point where the bare mass m BC vanishes. We would like in particular to apply diagrammatic expansion of the effective action (30). For this, it is useful to express the Ising part of the action in term of Nambu-Gorkov representation of the fermions [START_REF] Mattuck | A Guide to Feynman Diagrams in the Many-body Problem[END_REF][START_REF] Tsvelik | Quantum Field Theory in Condensed Matter Physics[END_REF], using the two-component objects

Φ k =   c k c-k   , Φk = (c k , c -k ). (37)
Formally, the Green functions can be defined within this representation by 2

× 2 matrices Ĝ(k) = Φ k Φk τ 3 , (38)
where τ 3 is the Pauli matrix [ 11] showing comparison between critical line defined by the vanishing mass (35) (plain red line) and numerical results from Monte Carlo simulations. The black filled dots are from figure 1, da Silva et al. [ 19] (Wang-Landau method). The cross symbol indicates the tricritical point identified by [ 11] using a Hartree-Fock-Bogoliubov approximation for the quartic part (33) of the effective action. The blue diamond symbols are from Ref. [ 20], the magenta triangles from Ref. [ 21], and the green squares from Ref. [ 22] (see also Table 1 for other numerical values at ∆ 0 = 0).

τ 3 =   1 0 0 -1   . (39) 14 -8 -6 -4
The unperturbed part of the Green function Ĝ0 is evaluted using the elements of the non diagonal but quadratic Ising action (32):

Ĝ0 (k) = -1 |m k | 2 + 4t 2 k x k y   mk 2itk y 2itk x m k   , (40)
where the momentum-dependent mass is defined by

m k = m BC + it(t + 1)(k x -k y ).
The inverse is given by Ĝ-1

0 (k) =   -m k 2itk y 2itk x -mk   . ( 41
) q k ′′ k ′′ -q Vk ′′ ,k ′′ -q k ′ k ′ + q 1 Figure 2.
Representation of the interaction part of the potential. The blob represents the potential interaction with incoming vector k ′′ and outgoing k ′′q.

With this representation and the unperturbed Green function, we can write the Ising part as

S Ising = k∈S Φk τ 3 Ĝ-1 0 (k)Φ k , (42)
where the set S contains half the momenta of the Brillouin zone. It is defined by the rule that if k ∈ S, then -k does not belong to S. The interaction part can be put, after some algebra, into the following form

S int = -g 0 1 4L 2 k ′ ,k ′′ ,q Φk"-q τ 3 Vk"-q,k" Φ k" Φk ′ +q τ 3 Φ k ′ , (43)
where the sum is not restricted to the ensemble S. We define the potential matrix V by

Vk"-q,k" =

  V k",k"-q 0 0 V k"-q,k"   . (44)
The two diagonal elements of this matrix are not equal since V k,k ′ is not symmetric by exchange of the two momenta k and k ′ except when k = k ′ .

We now perform a diagrammatic expansion with respect with g 0 of the perturbed Green function Ĝ(k) which will allow us to compute the corrections to the mass,

(a) q k k Vk,k (b) q k k Vq,q (c)
q k + q k k Vk,k+q (d)
q k -q k k Vk-q,k 1 Figure 3.
The four diagrams appearing at the lowest order in g 0 . Only diagram (b) contributes to the mass in the low momentum limit k → 0.

Γ(k) = Ĝ(k) -1 = Ĝ0 (k) -1 -Σ(k). (45)
The first terms contributing to the self-energy are given in figure 3:

Σ(k) = -g 0 1 4L 2 q Vk,k+q Ĝ0 (k + q) + Ĝ0 (k -q) Vk-q,k -g 0 1 4L 2 q
Vk,k Tr Ĝ0 (q) + Vq,q Tr Ĝ0 (q) .

The renormalized mass m R is given in the limit when k is zero by the diagonal components of the inverse-propagator Γ 11 (0) = Γ 22 (0). In this limit, only one diagram is not vanishing, which corresponds to the diagram (b) of figure (3):

m R = Γ 11 (0) = m BC + g 0 1 4L 2 q V q,q Tr Ĝ0 (q), = m BC -m BC × g 0 1 2L 2 q V q,q m 2 BC + t 2 (1 + t) 2 (q x -q y ) 2 + 4t 2 q x q y . (47)
The last sum over q can be evaluated in the continuous limit L → ∞. Setting q = 2π( m L , n L ), we define for large L the two following integrals

I 1 (m BC ) = g 0 8π 2 2π 0 2π 0 dq x dq y q x q y m 2 BC + t 2 (1 + t) 2 (q x -q y ) 2 + 4t 2 q x q y , I 2 (m BC ) = g 0 8π 2 2π 0 2π 0 dq x dq y q 2 x + q 2 y m 2 BC + t 2 (1 + t) 2 (q x -q y ) 2 + 4t 2 q x q y . (48)
These two quantities are finite when m BC vanishes. To see why, we can consider polar coordinates q x = q cos θ and q y = q sin θ, so that, near the origin q = 0 the second integral for example behaves like

I 2 (m BC ) ∝ g 0 8π 2 t 2 (1 + t) 2 0 2π 0 q dq dθ 1 {m BC /qt(1 + t)} 2 + 1 + {-1 + 2/(1 + t) 2 } sin(2θ) .( 49
)
When m BC = 0, this integral is finite since there is no singularity in the denominator.

Indeed when θ = ±π/4 the modulus of the term -1 + 2/(1 + t) 2 is strictly less than one on the critical line. This would not be the case if the last term 4t 2 q x q y in the denominator and coming from the off-diagonal part of the Green function was absent. In this case the integrals would be singular in the limit of small mass m BC , the denominator would instead be equal to {m BC /qt(1 + t)} 2 + 1sin(2θ), and the singular part would behave like 1/|m BC | by a simple scaling argument, which would cancel the other mass term m BC in front of the integrals (47). Then the renormalized mass would be shifted by a finite quantity, as well as the critical line.

Here the renormalization only concerns the total coefficient of m BC and this does not affect the critical line location:

m R ≃ m BC 1 -αI 1 (0) + α ′ I 2 (0) . (50)
A plot of the positive ratio m R /m BC evaluated at criticality m BC = 0 as function of ∆ 0 is given in figure 4. It is close to unity for almost all values of ∆ 0 .

This analysis shows that corrections to critical temperatures are indeed small and are of order higher than g 0 . A further analysis should be carry out to obtain a finite shift by considering more complex diagrams. It is also supported by analytical values of m BC = 0 which are close to numerical results reported in figure 1.

Extension to other spin-S models: generalization of the bare mass

The previous bare mass computed in the BC model, m BC , allowed us to obtain a precise description of the second-order critical line in the phase diagram. This was obtained by taking the limit of low momentum in the effective action, see (30). This can be generalized for any value of the spin S, in particular for higher values of S.

The equations obtained in section 2, equations ( 16) and ( 17), are general for they represent the fermionization of any spins-S model.

The construction of the fermionic action is however not an easy task, unlike the BC model which is a simpler case, but we expect to be able to extract a bare mass associated to non kinetic terms, or term involving derivatives with respect with space variables. At first approximation, we assume that the partition function and the free energy are singular in the low momentum limit when this bare mass vanishes. In the continuum limit, the c's coefficients defined by relations (13) can be rewritten using formal derivatives, such as c 1 mn = x q (ā q mn -∂ x āq mn ), etc... The derivatives contribute only to the kinetic energy and not to the bare mass. Keeping the first terms of the expansion, c 1 mn ≃ x q āq mn (as well as for the other coefficients c's), the weights W mn become uncoupled in the sense they contain variables depending only on local site (m, n) and we define the mass m S as: m S ≡ u 2 0 q α=1 dā α mn da α mn d bα mn db α mn e a α mn āα mn +b α mn bα mn W mn , with c 1 mn ≃ x q āq mn , • • • , c q mn ≃ x 1 ā1 mn and c q+1 mn ≃ x q bq mn , • • • , c 2q mn ≃ x 1 b1 mn . The integral can be evaluated exactly by noticing for example that the arguments of the exponential b α mn bα mn can be combined with a a α mn (x α āα mn ) that appears in some of the q (2k) k=1...2q products to give a contribution x α . Indeed using the Grassmann integration rules da.a = 1 and da. Also a term a α mn āα mn can be combined with b α mn (x α bα mn ) to give the same contribution. Since the q (2k) are ordered, there are also signs to take into account and coming from the fact the variables c α mn have to be moved in the correct order before integration. We obtain after some algebra the general relation

m S = 2q k=0 α k R k , (51)
where we have define the following quantities with initial condition

R 0 = u 2 0 , R k = k l=0 u l u k-l σ(l, k -l), (52)
and σ(k, l) = 1 if k and l are both even, and σ(k, l) = -1 otherwise. We can apply this result to different cases to check the validity of this relation. For the Ising model (S = 1/2, q = 1) u 0 = ch(K/4) and u 1 = 4sh(K/4), we obtain

m 1/2 = 2 cosh(∆/4)(u 2 0 -u 0 u 1 /2 -u 2 1 /16), or m 1/2 = 2e ∆/4 [1 -sh(K/2)], (53)
which vanishes at the Ising critical temperature T c ≃ 0.567 296 or with the normalization t c ≡ T c /S 2 = 2.269 185, which is independent, as expected, of ∆ 0 . For the Blume-Capel model (S = 1, q = 2) we have

m 1 = 1 + 2 exp(∆)(1 -2u 1 + 2u 2 -u 2 1 - 2u 1 u 2 + u 2
2 ), or more explicitly 1.203 [START_REF] Butera | [END_REF] 0.915 [START_REF] Butera | [END_REF]29] This mass is directly proportionnal to the mass m BC found in the previous section.

m 1 = 1 + 2e ∆ [1 -sh(2K)]. (54)
Indeed, we have the relation

m BC = g 0 m 1 (55)
and therefore both masses vanish on the same line of critical points. The coefficient g 0 comes from a global rescaling of the Grassmann variables in the original weights W mn which leads to the coefficient g -L 2 0 in the BC function partition (19) and (22), instead of the coefficient (u 0 ) 2L 2 = 1 in front of [START_REF] Plechko | [END_REF]. For ∆ 0 = 0 we find in particular that t c = 2/arcsinh(3/2) ≃ 1.673 971 (see table 1).

The other masses are deduced by iteration of formula (51) and (52). For higher values of spin S, we find:

m 3/2 = 2e ∆/4

Conclusion

In this review paper, we have presented a method which tries to operate a correspondence between classical spin models and fermionic systems. We have extended Plecho's method [START_REF] Plechko | [END_REF]10] based on Ising model to generalized spin-S systems. The method is based on the projection onto q polynomial components, q depending specifically on the value of spin S, of Boltzmann local weights given by equation (2).

Then the introduction of 2q pairs of Grassmann variables per site and the use of special symmetries such as mirror and associative symmetries in 2D for Grassmannian objects allows us to perform exactly the sum over the spin variables. This gives a representation of spin-S models in term of fermionic multiple integrals [START_REF] Plechko | [END_REF]. Effective actions can in principle be deduced from this representation. We have shown that such action can be built exactly for the Blume Capel model S = 1 (30) and the bare mass (35) gives accurate description of the second-order critical line. We have seen that there is no shift of this mass due to the effect of quartic potential of the effective theory at the lowest order expansion in the coupling parameter g 0 , implying that corrections to the critical temperature may be indeed small. This quartic potential is however responsible for the presence of a tricritical point, rendering the second order line instable by changing the sign of the stiffness coefficient or making the free-fermion spectrum itself instable. For general spin-S model, the bare mass can also be generalized and calculated directly in the low momentum limit (51) without knowing the full effective action, and still gives accurate description of second-order critical points even in the limit of the continuous Ising model.

c

  M. Clusel, J.-Y. Fortin, 1 MC-JYF

Figure 1 .

 1 Figure 1. (color online) Figure taken from reference[ 11] showing comparison

  i.e. the shift of the critical temperature, by identification of the diagonal elements of the inverse propagator Γ(k) = Ĝ(k) -1 . The graphical representation of the matrix potential in term of diagram is displayed in figure 2. To do so we follow formally the Feynman rules which lead to the Dyson equation of the inverse-propagator in term of the self-energy Σ(k):

Figure 4 .

 4 Figure 4. Plot of the coefficient renormalizing the BC bare mass as function of ∆ 0 on the critical line. The values are close to unity except near the region of the terminating point ∆ 0 = 2 where g 0 is not small.

[ 1 -

 1 sh(K/2)] + 2e 9∆/4 [1sh(9K/2)] + 2e ∆/4 [1sh(K/2)], m 2 = 1 + 2e ∆ [1sh(2K)] + 2e 4∆ [1sh(8K)], m 5/2 = 2e 25∆/4 [1sh(25K/2)] + 2e 9∆/4 [1sh(9K/2)] + 2e ∆/4 [1sh(K/2)], m 3 = 1 + 2e 9∆ [1sh(18K)] + 2e 4∆ [1sh(8K)] + 2e ∆ [1sh(2K)].(56)

Table 1 .

 1 Comparison of critical temperature values at ∆ 0 = 0, solutions of equation (57), with other methods found in bibliography.

	Spin S	S = 1/2	S = 1	S = 3/2	S = 2
	q	1	2	5	6
	T c	0.567 296	1.673 971	3.277 561	5.351 248
	t c = T c /S 2 2.269 185	1.673 971	1.456 694	1.337 812
	Refs.	2.269 [ 25] 1.689 [ 26], 1.695 [ 19] 1.461 [ 27, 25, 28] 1.336 [ 25]
			1.694 [ 25], 1.681 [ 27]
	Spin S q T c t c = T c /S 2 1.262 542 S = 5/2 11 7.890 888	S = 3 12 10.894 806 1.210 534	S = ∞ ∞ ∞ 0.925 148
	Refs.	1.257 [ 25]

For general spin S, we can extend the previous results to the formula

This is a simple result giving a precise location of the second-order critical lines by solving the equation of the bare mass m S = 0. We give for comparison tabulated values of t c at ∆ 0 = 0 in table 1 for different values of S, and references to numerical results (Monte-Carlo simulations, high-temperature expansions) given in the literature. In general, the agreement is good up to 1% in most cases. For half-integer values of S, the models possess an asymptote in the (T /S 2 , ∆ 0 ) plane. Indeed, for S = 3/2 for example, the equation given in (56) predicts the solution

which is bounded by t c = 2/9 log(1 + √ 2) = 0.252 131 below which there is no continuation of the second-order critical line. In the limit of large integer S, the model defined in equation ( 1) is described by a continuous variable -1 < x mn = S mn /S < 1 and is called continuous Ising model. We can still obtain a finite value of the critical line by taking the asymptotic value of equation ( 57)

and in particular for ∆ 0 = 0, we have the following expansion for large

with a(t) = 2 -2 2/t √ 2/t 0 sh(x 2 ) dx. We observe that the rescaled mass m S /S vanishes in this case when t c = 0.925 148, in good agreement with numerical works for this model [START_REF] Butera | [END_REF]29], and it is worth noting that equation (59) also possesses a non trivial solution at t = 0 which is simply given by ∆ 0 = 4/ √ 3 = 2.309 401. This value is different from the value 2 expected for all finite S models [ 8]. It can be suggested that there also exists a tricritical point before this non physical value is reached.


Denis Michel
email: denis.michel@live.fr

Resolution of twin paradox controversies

The thought experiment called the clock or twin paradox proposed in the context of the theory of relativity, is the most emblematic illustration that different worldlines have different proper times. Its solution predicting an absolute time shift between the inertial and non-inertial twins in the ratio of a Lorentz factor, still raises some reticences when presented as an unilateral application of time dilation of uniform translation, otherwise reciprocal in special relativity. The alleged solutions based on time dilation or acceleration are questioned and preferably replaced by the one provided by the ideal light clocks of Einstein that demonstrate a break in temporal reciprocity independent of the choice of viewpoint at rest.

Introduction

As early as his seminal 1905 special relativity paper, Einstein raised at the end of his section 4, the issue of a traveling clock returning from a round trip, concluding that it will lag by 1 2 v 2 c 2 on its stationary counterpart [START_REF] Einstein | Zur Elektrodynamik bewegter Körper (On the electrodynamics of moving bodies)[END_REF]. The principle of the absolute time shift was stated. The issue is important because it has in fact little to do with the rest of his paper on special relativity. It is no longer a question of the symmetrical relativistic time dilation between two observers-emitters in uniform motion with respect to each other. In the round trip, the time distortion effect is not reciprocal. The literature and online forums are full of different opinions on this experiment, the most popular version of which is called the Langevin twin experiment whose principle was developed for any beings [START_REF] Langevin | L'évolution de l'espace et du temps[END_REF] and then adapted to twins [START_REF] Weyl | Vorlesungen über Allgemeine Relativitätstheorie[END_REF]. Some authors claim that this question has been solved for a long time in the the context of special relativity while others claim that only the use of the tools of general relativity can solve it. In the history of physics, this experiment is atypical in the number of solutions proposed and the animosity of the debates it has generated. An anthology of the arguments put forward can be found in [START_REF]The Twin Paradox[END_REF][START_REF]The Twin Paradox[END_REF][START_REF] Mathpages | The Doppler Twins[END_REF][START_REF]Twin paradox link wikipedia[END_REF][START_REF]Twin paradox[END_REF][START_REF] Rodrigues | The meaning of time in the theory of relativity and Einstein's later view of the Twin Paradox ?[END_REF][START_REF] Sachs | On Einstein's later view of the twin paradox[END_REF][START_REF] Sachs | Response to Rodrigues and Rosa on the twin paradox[END_REF][START_REF] Grandou | On the Ingredients of the Twin Paradox[END_REF][START_REF] Gamboa | The twin paradox : the role of acceleration[END_REF][START_REF] Mohazzabi | Has the twin paradox really been resolved ?[END_REF], [START_REF]Twin paradox report[END_REF][START_REF] Mcdonald | The clock paradox and accelerators[END_REF] and the many references therein. Einstein himself repeatedly went back on his initial conclusion about the traveling clock, which likely reflects doubts. In 1918, he proposed that finally this question should rather fall under general relativity (which he had not yet developed in 1905) because accelerations generate the equivalent of a gravitational field, but without formulating or quantifying his new proposal. The thought experiment of differential aging then became so toxic that eminent relativists, such as Dingle, lost their scientific credibility. Editors fearing for the reputation of their journals censored the subject by rejecting manuscripts without reading them, which may have been wise because many articles dealing with the twin paradox have been attacks against the theory of relativity. There are a few mentions in journals aimed at the general public and repeating the explanation related to time dilation [START_REF] Lasky | Time and the twin paradox[END_REF]. Then the censorship became self-censorship. Few recognized physicists venture into the discussion today. When some academic authors consider that the denial of the standard solution results from the apparent paradox between the special relativity theory and the common sense of non-experts in relativity, some opponents of this result consider that the paradox is rather between the official solution and the well understood special relativity theory. The major conflicts opposed (1) authors considering the principle reciprocity of time dilation during the phases of uniform relative motion, as a fundamental of the special relativity theory and (2) those who, while claiming to defend special relativity, consider that the rule of reciprocity can be dispensed with because of the asymmetry of the astronaut's turn around. Einstein also used such an argument in 1911 when he wrote : accelerations are not relevant to quantify the time difference between the two clocks, but their presence nevertheless causes the moving clock to slow down and not the fixed clock [START_REF] Einstein | Die Relativitätstheorie[END_REF]. And finally (3) the supporters of general relativity, probably uncomfortable with the loss of reciprocity in special relativity. Einstein was not far from adopting this position in 1918 when he went back on his 1911 hypothesis by admitting that special relativity time dilation was symmetric for twins [START_REF] Einstein | Dialog über Einwände gegen die Relativitätstheorie[END_REF], which had led him to assume and that it was therefore necessary to invoke, asymmetrically, general relativistic gravitational time dilation during the brief periods of acceleration in order to justify asymmetric aging. The present study focuses on solutions reconciling an absolute temporal shift with a strict respect of the principle of reciprocity in relative uniform motion. Let us begin by describing below the common treatment of the twins experiment which has caused much misunderstanding and has probably been the origin of the term paradox. Then a much more consensual explanation will be detailed, using a powerful conceptual tool : the perfect clock designed by Einstein, consisting of a light tube that will simply mark the time units elapsed for each twin.

The majority treatment of the twin experiment

Differential aging should not be confused with the time dilation of special relativity which is a perfectly symmetrical phenomenon since it is impossible to attribute a relative mobility between two inert clocks to one rather than to the other. Differential aging is absolute and measurable between two clocks which, between two points of encounter, have undergone different experiences including changes of speed. One of the major causes of the reluctance to accept the official result of this experiment could be that the time elapsed between the meeting points of the twins are in a ratio corresponding to the value of the time dilation (γ) of special relativity whereas these phenomena are not clearly related. Even worse, a frequent resolution of this experiment look like an asymmetric application of the time dilation of special relativity, which is basically symmetric. Let us recall this experiment. The traveling twin will be noted A, the inert twin will be noted B and the point of U-turn, as well as a third inert observer posted at this point, will be noted C (Fig. 1A). For simplicity and without loss of generality, the turnaround will be assimilated to an elastic rebound, i.e. a simple inversion of the velocity vector at point C, without modification of its magnitude (non-inertial phase of infinite acceleration and zero duration). The conclusion generally given is that A and B are no longer the same age, A being younger than B and C on arrival, although the reason given is variable. The debates on the subject have agreed on two major consensuses : (1) when the traveller finds his brother on his return, he is younger than him and (2) the two brothers' journeys have been dissymmetrical because, unlike his sedentary brother, the traveller had a non-inertial experience because of these multiple accelerations, at takeoff, at landing and especially when he turns back. Fig. 1C is logically forbidden because it gives the misleading illusion that there is a reference frame for the rocket. The complete path of A is in fact chimerical from the point of view of special relativity. The reference frames of the outward and return journey cannot be placed in the straight extension of each other and should be noted more rigorously as different states, for instance A O during the outward trip and A R during the return trip. But all the authors do not give the same role to this dissymmetry. For some, the solution lies entirely in the acceleration and must involve general relativity. But this proposal in which the time lag on arrival would entirely originate from the one-off bounce event, can be easily refuted because the non-inertial phases can be made as negligible as one wishes by lengthening the fraction of the trip made in uniform motion. To attribute the age difference to the accelerations would be to abstract from the duration of the journey, which no author envisages in the calculation of differential aging. A time lag concentrated in a single point could not explain a result depending on the length of the trip. The assumption that acceleration is the cause of time lag was attractive because of its asymmetry, but as we shall see, other asymmetries exist. Finally, acceleration cannot be invoked as an intrinsic cause of differential aging as perfect atomic clocks are unaffected by acceleration. Once the necessity of recourse to general relativity has been eliminated, the half-turn has been put forward to justify that only the traveler moves and that therefore the time dilation of special relativity must apply exclusively to him. In other words, since the diagram in Fig. 1B is forbidden, the effects of special relativity will only be deduced from the diagram in Fig. 1A. The shortest modeling of this decison is to use length contraction. If the total distance traveled during the trip is L, then for the brother who stayed at home, the duration of the trip is

∆t B = L v (1a)
while taking into account the shortening of L for the traveller, the duration of his trip is only Of course, the diagram in Fig. 1B is forbidden, but other diagrams can be represented instead, for example from the point of view of the travelling twin at rest during either the outward or the return journey only. So the prohibition of Fig. 1B does not justify in any way the absence of time dilation of A for B. The results of special relativity during uniform relative translation are in essence reciprocal so that B is also subject to time dilation for A. The representation of a round trip of the earth with respect to the rocket (Fig. 1C) which would have allowed to invert the equations Eq.(1a) and Eq.(1b), is forbidden. This prohibition seems contradictory with the absence of an ether which deprives us of a fixed reference frame during the phases of uniform motion. As pointed out by Poincaré [START_REF] Poincaré | Le principe de Relativité. La dynamique de l'électron[END_REF], obviously a uniform relative motion between two objects is by definition relative and cannot be specifically attributed to one or the other. Whether one of the two intends to turn back in the future or has turned back in the past makes no difference, because the realization of this intention does not belong to the same plane of simultaneity as the uniform motion. Hence, the decision not to assign the contraction of Eq.(1b) to B is not justifiable. Two arguments are presented below to further grasp the necessity of reciprocity during the outward as well as return journeys. During the outward journey, imagine that the traveling twin A1 is followed by a second astronaut A2, at a distance D from him in his rest frame. The interval between A1 and A2 is perceived by B as D 1 -v 2 c 2 because for B, when A2 reaches it, A1 has not yet arrived at C. A rule in special relativity is that when two distance intervals cross each other at constant velocity, each one keeps its length seen internally and sees the other one contracted. We can also use an elegant method called here the moving observation frame, whose relevance and efficacy are shown in the appendix A. In Fig. 2, the moving observation frame U is in uniform motion with respect to A and B, with a velocity u relative to B. Its vector is oriented like the velocity vector of A and its magnitude is fixed very precisely at

∆t A = L v 1 - v 2 c 2 (1b)
u = c 2 v 1 -1 - v 2 c 2 (2)
in such a way that v = u ⊕ u. Note that like A, U makes a half turn so that the rest frames of U shown in the top and bottom panels of Fig. 2 are different and written U and Ũ respectively.

• During the outward journey, U sees A and B moving away from each other with the same velocity modulus u and their apparent time is changed by exactly the same factor.

• During the return trip, Ũ sees A and B approaching each other, still with the same uniform velocity modulus u and sees their apparent time modified once again by the same factor. Writing ∆t mov BU the duration of B measured from the rest frame of U and ∆t B the proper duration for B, we have

∆t mov BU ∆t U = ∆t mov AU ∆t U = ∆t mov B Ũ ∆t Ũ = ∆t mov A Ũ ∆t Ũ = 1 1 -u 2 c 2 (3)
We do not recover the asymmetry of Eq.( 1). The moving frame of Fig. 2 also allows us to reconsider the importance of the half-turn, which for many authors is supposed to bring this experiment into the field of general relativity.

Figure 2 Conservative modification of the velocities by the method of the moving observation frame (U , the rectangle with rounded corners). By giving this frame of reference an appropriate uniform velocity, it is possible to assign identical velocity magnitudes to A and B while maintaining the relativistic equivalence with the classical statement of the twin experiment.

The defenders of a general relativity-based solution put forward the shaking associated with the U-turn, with the fall of the pots and pans hanging on the wall, which would not have taken place in uniform motion. But on reflection, even the half-turn is a question of point of view. Not only does the moving frame distribute the velocities equally between A and B, but also, by inversion of its velocity vector, the switch between the point of views of U and Ũ also distributes the half-turn between A and B. The half-turn attributed exclusively to A in the usual description without moving frame, is thus partly transferred to the half-turn of a third actor : U . Therefore it becomes difficult to impute the same gravitational upheavals to A, which removes the need for a solution based on general relativity. Consider a variant experiment in which A does not make its half-turn but continues its uniform motion, and that the return trip is made by another astronaut A of uniform velocity -v who crosses A precisely at the level of the planet C. At the crossing point, A turns off his flashes while A turns them on. The result of this experiment would be the same as the classical one, but without acceleration and without the noise of broken dishes in the ship. The reference frames of A and A are clearly different since if they coexisted, their relative velocity would be v ⊕ v.

In these conditions, we can understand the conflict that arose between the protagonists of this debate. The supporters of the official solution accuse their opponents of not understanding special relativity, whereas the latters precisely rely on special relativity to refute the official solution. Faced with the poor arguments found in the literature on the subject, it is interesting to consider the opinion of the famous physicist Richard Feynman, known for his pedagogy and his physical lucidity. Unfortunately, his development on this subject does not tell us anything new because it uses the typical presentation in three points : (1) The result : through special relativity (time dilation), the sedentary person sees his brother evolving more slowly, (2) The traditional question : but shouldn't this perception be reciprocal ? [START_REF] Weyl | Vorlesungen über Allgemeine Relativitätstheorie[END_REF] The answer to this question : no because only the traveler undergoes an upheaval at the time of the half-turn [START_REF] Feynman | The Feynman lectures in physics[END_REF]. It comes back to the idea that although it is not used in the treatment, the non-inertial U-turn just allows us to ignore the reciprocity of special relativity during the inertial phases of the trip. Fortunately, the perfect and universal clock that is Einstein's light tube will give a more rational explanation.

Illustration of the main results of special relativity with the light clock

Einstein's universal light-clock, which is very instructive in illustrating the rudiments of special relativity, also indisputably resolves the much-discussed issue of differential aging. Let us recall two basic axioms of relativity :

• In vacuum, the speed of light is finite and constant for all observers, as anticipated in the fundamental relations of Maxwell. Poincaré has shown that the transformations of relativity leave Maxwell's relations unchanged.

• A uniform relative motion (without accelerations) between two objects is perfectly reciprocal, i.e. it is impossible to attribute the motion to one or the other.

Let's see the consequences of these rules on the observation of a clock designed by Einstein.

The light tube

The light tube is a simple tube (easier to describe than to realize in practice) with perfectly parallel mirrors at its ends. The ticking of this rudimentary clock is the incessant bouncing of the light between the two mirrors. The unit of time is the duration of a return trip of the light between the mirrors. The fundamental concepts of special relativity can be deduced from experiments consisting in placing this tube on a cart in uniform motion with respect to the ground. We will first place the tube perpendicularly to the path of the cart, and then extended in the direction of the cart's motion.

Tube vertical on the cart

The tube is placed vertically on a carriage in uniform motion with respect to the ground, i.e. perpendicular to its path. For an observer placed on the cart (Fig. 3A), the tube appears fixed and the light travel time ∆t is just determined by the tube length c∆t. For an observer standing on the ground outside the cart, the light path appears oblique and therefore longer (Fig. 3B), with a length c∆t mov appearing modified by the movement of the tube. Since the tube has a unique size for both observers, either the travel time of the light is not the same (∆t mov > ∆t), or the speed of the light is modified (c mov > c). But as experience and reasoning have rejected this second possibility, there remains only that of the time. There are several approaches to deal with this scheme. As the tube moves during the path of the light, the light will have traveled a greater distance, but as the speed of light c is constant, he concludes that it is the duration of its journey that has increased. The Pythagorean theorem tells us by how much :

(c∆t mov ) 2 = (c∆t) 2 + (v∆t mov ) 2 (4a) giving ∆t mov ∆t = 1 1 -v 2 c 2 (4b)
Some could prefer elementary trigonometry using the angle θ of Fig. 3C :

sin θ = c∆t c∆t mov = ∆t ∆t mov (5a) cos θ = v∆t mov c∆t mov = v c (5b)
which gives, according to the relation sin

2 θ + cos 2 θ = 1, ∆t mov ∆t = 1 1 -v 2 c 2 (5c)
The durations read on a clock appearing in motion (perceived time) are longer than those read on the same clock appearing fixed (proper time).

Tube lying longitudinally on the cart

While in the previous experiment the tube was perpendicular to the path of the carriage, let us now place it lying in the direction of travel. The tube still appears motionless to an observer on the carriage for whom the path of light between the mirrors is the same in both directions (Fig. 4A). It is not so simple for the observer on the ground, for whom the intuitive reasoning is the following. The light going from the left mirror to the right mirror advances in the same direction as the carriage, so during its journey, the right mirror moves away as it advances. The rate of closure between the light and the mirrors is therefore slowed down to c -v. On the contrary, for the light that leaves the right mirror and returns to the left mirror, the latter comes to meet it, which increases its rate of closure to c + v. When keeping the speed of light constant, this reasoning leads to modify the available adjustment variables : the travel time of the light ∆t and the length of the tube L, so that for the observer on the carriage,

∆t = L 1 c + 1 c = 2L c (6a)
while for the outside observer seeing the mobile cart,

∆t mov = L mov 1 c -v + 1 c + v = L mov 2c c 2 -v 2 (6b)
By relating these two results, we find

L mov L = ∆t mov ∆t 1 - v 2 c 2 (7)
and by replacing in this equation the ratio of the durations by its value given by Eq.(4b), we obtain

L mov L = 1 - v 2 c 2 (8)
The perceived length of an object appearing in motion is shorter than that of the same object appearing fixed (proper length). This is the famous contraction of the lengths of special relativity in the direction of displacement (conventionally the x axis).

How the light tube illuminates the notion of spacetime

Euclidean and relativistic spacetime coordinates

Fig. 5 illustrates the consequences of the finite propagation speed of light. The classical view of panel A is the one that would prevail if the speed of light were infinite, i.e. if we were informed "in real time" of the exact location of everything. The diagram has the traditional coordinates of Newtonian motion : x the spatial dimension and t the temporal dimension. Note however that to anticipate the comparison with panel B, these axes are swapped compared to traditional representations where t is usually on the abscissa. The light tube shown on the left is fixed with respect to the observer while the tube on the left is moving at constant speed v which is the derivative of x with respect to t. This Newtonian view implies that the observer can see all parts of the light tube simultaneously since their images reach him together. Fig. 5B corresponds to the most common representation of special relativity : the Minkowski diagram. A major difference with the previous diagram is that the time axis is multiplied by the speed of light c to become ct and acquire the same dimension as x, thus defining a new notion of spacetime where the velocity of the tube will be more simply defined as the ratio v/c noted β which cannot exceed 1. As the speed of light is finite, the light tube and the light move together, which generates a pattern where the light going back and forth between the mirrors makes a zig-zag path in spacetime, already glimpsed on Fig. 3. A remarkable feature of this type of diagram is that the light paths are necessarily represented as oblique lines, at 45 • for all observers. A more subtle consequence of the finite speed of light is that the image of the back of the tube arrives at the observer before that of the front. Consequently the front is shifted a little towards the future, which causes the tilt of the moving tube on the diagram, with widened spacing between the graduations of the x coordinate with respect to those of x. The slope of this inclination corresponds to that of the x axis, is called the "simultaneity" line, along which all the parts of the moving objects are perfectly contemporaneous, i.e. exist at the "same present". With respect to the vertical ct axis of the frame of reference at rest, the tick-tock and tock-tick spaces are not of the same length, but if we divide the tick-tock-tick space by the tilt line of the tube, then the equality of the tick-tock and tock-tick intervals is restored. The lengthening of the inclined tube on the diagram follows the spacing of the graduations on the x axis and therefore its length remains unchanged in terms of number of units. However, by projecting the image of the tube on the x axis of the fixed observer, the tube is shortened from a length L when it was seen stationary, to L = L 1 -β 2 when it is seen moving. The Minkowski diagram associated to the light tube allows to clarify the most popular consequence of special relativity theory on time. The light tube is indeed a perfect clock whose elementary beat is a round trip (tick-tock-tick) of the light between the two mirrors. As the light paths are necessarily straight line segments at 45 • , it is obvious on Fig. 5B that the tick-tock-tick path is minimal for the immobile tube and is all the longer as the speed is high, i.e. as the slope of the trajectory of the mobile tube is strong. This dilation of the tick-tock-ticks could be only apparent because the oblique trajectory of the tube seems itself longer, but the tick-tock-ticks lengthen more than the path, as we will show by relating the two.

The geometry of the special relativistic spacetime

The light tube of Fig. 5B is perceived differently by observers of the reference frames (x, ct) and (x , ct ) but it is nevertheless unique. Precisely the rule underlying the Minkowski diagram is a rule of invariance that can be expressed as follows, to describe an interval between two same points of spacetime a and b

(ct b -ct a ) 2 -(x b -x a ) 2 = (ct b -ct a ) 2 -(x b -x a ) 2 (9a)
or, by adding the two missing spatial dimensions,

(ct b -ct a ) 2 -[(x b -x a ) 2 + (y b -y a ) 2 + (z b -z a ) 2 ] = (ct b -ct a ) 2 -[(x b -x a ) 2 + (y b -y a ) 2 + (z b -z a ) 2 ] (9b)
This spacetime interval is simply defined by

∆s 2 = (c∆t) 2 -d 2 (9c)
Its invariance allows us to find all the results of special relativity. Let's take the example of time dilation by describing again the scheme of the Fig. 1. The oblique (hypotenuse) and vertical paths of light start from the same point and arrive at the same point, so that this common interval (∆s) must reconcile the point of view of the observer on the cart, for whom the clock is stationary, and that of the outside observer for whom there is an additional translation

∆s 2 = (c∆t) 2 = (c∆t mov ) 2 -∆x 2 mov = (c∆t mov ) 2 -(v∆t mov ) 2 (10a) giving ∆t mov ∆t = 1 1 -v 2 c 2 (10b)
The invariance of ∆s of Eq.9 results from the Lorentz transformations which can be constructed in several ways : (1) by the formal description of the consequences of the axioms of finiteness and constancy of the speed of light [START_REF] Einstein | Zur Elektrodynamik bewegter Körper (On the electrodynamics of moving bodies)[END_REF] (2) from the postulate that they have the mathematical property of a group (If these transformations work between two objects A and B and between two objects B and C, then they will necessarily work between A and C). ( 3) They can also be deduced in a reverse way from a geometric analysis of the Minkowski spacetime diagram. Note that the intervals ∆s are not directly apparent on the Minkowski diagram, because it is a hyperbolic distance y 2 -x 2 which is not the traditional Euclidean distance y 2 + x 2 . The Lorentz transformations can be compared to hyperbolic rotations, different from the usual circular rotations. In this respect, Minkowski diagrams can appear confusing with respect to time units. By hyperbolic normalization, the units (spacing between the graduations) of the frame (ct , x ) are dilated with respect to those of (ct, x) in the same proportion for ct and x , which is fortunate because a relative velocity is necessarily reciprocal (v = ∆x/∆t = ∆x /∆t ). The angle between the axes x and x is therefore the same as that between the angles ct and ct (arctan β). The scale factor of the axes is given by the correspondence between the spacetime intervals such that ∆s 2 = (c∆t) 2 -∆x 2 = (∆x sin arctan β + c∆t cos arctan β) Hence the apparent expansion of the (ct , x ) diagram units is,

∆s = ∆s 1 + β 2 1 -β 2 (11b)
This purely graphic dilation is not the real time dilation of special relativity, which is in fact given by the ratio between the intervals of the axes t and t included between parallel lines of simultaneity. We have geometrically between these lines

∆t = 1 + β 2 1 -β 2 ∆t cos arctan β = 1 + β 2 1 -β 2 ∆t 1 1 + β 2 = ∆t 1 -β 2 (12a)
Time dilation is therefore not so obvious to illustrate with Minkowski diagrams and these diagrams may be misleading. If we give to two observers in uniform motion the same synchronized clock during a common event, then the intervals between two successive ticks are exactly the same for both of them in Minkowski diagrams once we take into account the correction by the scaling factor that we just saw.

Counting the tick-tock-ticks

The light tube which has already proved very useful to easily find the laws of special relativity, is also powerful and pedagogical to count the time units defined in the exercise L-2 of [START_REF] Wheeler | Spacetime Physics[END_REF] and in [START_REF] Salgado | Visualizing proper-time in special selativity[END_REF]. The numbers of time units can be obtained in several ways but a more direct one presented in the appendix B, consists simply in counting the "tick-tock-tick" sequences contained in each path, by dividing the length of the path by the length of a tick-tock-tick interval measured on Minkowski diagrams. We can use the close-up diagrams of Fig. 5B and Fig. B1 to find the length of a tick-tock-tick, and the far-up diagrams of Fig. 6 to find the length of the sections performed at constant or zero speed. The ratios of these distances give the numbers of time units marked by the light tubes. For the traditional diagram of Fig. 6A,

N ac = D Lβ (13a)
N ab = N bc = D 2L 1 -β 2 β ( 13b
)
(appendix B). We are accustomed to calculations from the reference frame of the inert twin at rest, as for example in [START_REF] Lasky | Time and the twin paradox[END_REF][START_REF] Feynman | The Feynman lectures in physics[END_REF]. But this near-generalized choice appears arbitrary and insufficient for the purists who want to make sure that an absolute result does not depend on the point of view. Therefore, a more convincing proof of this result would be if it could also be obtained from the point of view of the traveler. Precisely, the tick-tock-tick method allows this. For the diagram of Fig. 6B established from the point of view of the traveler at rest during the outward journey, a new tick-tock-tick interval appears on this diagram for the the section bc, because the speed of the astronaut with respect to his outward journey is now v ⊕ v. But on the diagram the lengths of the sections of uniform motion vary in concert with these intervals (appendix B) in such a way that by making the ratio, their number is unchanged. The calculations of the ticks depend on the points of view but that in all the cases, the resulting ratio is always

N ab + N bc N ac = 1 -β 2 (14)
The identical light clocks of the twins, synchronized at the beginning and unfalsifiable, show undoubtedly that the numbers of ticks emitted on the planet of departure and in the rocket are different. In addition to the calculations detailed in the appendix B for the diagrams Fig. 6A and Fig. 6B, calculations from any other point of view all give the same result. One can start for instance from the point of view of a new traveler in Fig. 6D, with a speed u given by Eq.( 2), slower than the traveling twin. He would remain at mid-distance between him and the earth but would forget to turn around. From its rest frame, the earth and the traveling twin would move away from it at equal velocity magnitude, and then he would cross the path of the returning traveler. In all cases, tick-tock-tick intervals vary in concert with the paths, keeping their number constant. This resolution of the twin paradox is simpler than but analogous to that of the analysis of the lines of simultaneity which is more delicate to grasp. The lines of simultaneity represented by the inclination of the tube in Fig. 5b are valid for the outward journey but must be tilted inversely for the return journey. The transition between the two inclinations thus causes a "simultaneity gap" in the intersection of these lines on the ct axis, embodies the "time saved" by the travelling twin [START_REF]Twin paradox link wikipedia[END_REF][START_REF]The Twin Paradox[END_REF].

Numerical example

Take the case of a round trip at the speed of 0.6 c between two planets distant of 15 light minutes. The twins have identical light clocks of 3 cm length perfectly synchronized at the beginning and whose orientation is fixed with respect to the cosmos thanks to perfect Foucault gyroscopes. According to the previous formulas, between the separation and the reunion of the twins, the light tube of the inert twin marks D Lβ = 15 × 10 12 units of time, each one worth 2 × 10 -10 seconds, that is to say a 50 minutes duration. This is the commonly expected duration of a 30 minute trip light at 0.6c. By contrast, the traveling twin counts only 12 × 10 12 units of time at his light clock, and a total duration of 40 minutes.

The delicate use of time dilation to obtain this result

The widespread treatment of the twins experiment using time dilation leads to a result identical to the one stated above, describable with the famous gamma factor. This unfortunate identity has probably caused a long standing confusion, amplified by the fact that strangely en-ough, the result of the twins experiment was given without sufficient explanation in Einstein's article essentially devoted to special relativity. It is however essential to make a clear distinction between the two phenomena. Time dilation can exist continuously between two observers between two common events without causing any differential aging when they meet.

The time dilation of uniform motion is definitely reciprocal

The fundamental reciprocity of the special relativity theory

No one disputes the fact that the twin B left on the planet of departure is inert while his astronaut brother A is not. Nevertheless, this situation does not compromise the symmetry of the time dilations between A and B, as illustrated in the spacetime diagrams of Fig. 7A and Fig. 7B drawn in the rest frame of a slow astronaut running in the same direction as A and always present halfway between A and B thanks to its relative velocity previously determined for the mobile observation frame (Eq.( 2)). This third observer obviously understands that the relative time dilations A and B are strictly the same for A and B during the outward journey (Fig. 7A) as well as the return journey (Fig. 7B). Note however, that the direction of this observer differs between the two parts of the journey. Because of the prohibition of Fig. 1C, the usual interpretation of the twin experiment has been derived essentially from the interpretation of the remaining diagram (Fig. 1B), but in special relativity when two bodies A and B are in uniform translation with respect to each other, the value of the reciprocal time dilations (of A measured by B and of B measured by A) are equal, which means that their proper times also evolve in parallel. Let us limit the experiment to the outward journey before the half-turn, as represented in Fig. 7A. This diagram is just as exact as that of Fig. 1B, but seen from the rest frame of the astronaut U . The symmetry of the time dilation of special relativity obviously ensues from the symmetry of the Lorentz transformations. For the readers still unconvinced, let us place ourselves in the rest frame U of the slow astronaut who leaving the earth at the same time and in the same direction as A at the velocity u of Eq.( 2), and then called Ũ when he is back. The diagrams seen from the rest frame of this astronaut halfway between B and A are shown in Fig. 7. On Fig. 7C, the dizzy astronaut A O forgot to make his half-turn and doubles straight the path he just made. In this case, the meeting event c does not take place but there is no doubt for U that the proper times elapsed on the watches of A and B must be the same. The reciprocity of time dilation between uniformly moving bodies is sometimes misinterpreted as x < y and y > x, although it is in fact no way paradoxical. Calling ∆t mov AB the duration of A seen from the rest frame of B and ∆t B the proper duration of B, the symmetry between these diagrams reads

∆t mov AB ∆t B = ∆t mov BA ∆t A = 1 1 -v 2

Time dilation is unable in itself to generate differential aging

Let us imagine a variant twin experiment, where another astronaut A 2 takes the opposite path of A 1 (Fig. 8).

Both undergo the same non-inertial turnaround but because of the obvious symmetry of their trips, it is impossible to impute any differential aging on arrival between A 1 and A 2 . Yet, throughout their journey, they were subject to time dilation due to their constant relative velocity v⊕v. Accordingly, they would have perceived increased Doppler effects from each other, with a dilated period during their spacing of

T L = T 0 1 + v c / 1 -v c
and a shorthened period during their approach of T S = T 0 1 -v c / 1 + v c . We see that there is no reason why a differential aging between two bodies corresponds to the traditional phenomenon of time dilation between these bodies.

The hidden internal time dilation

The uniform relative motion between two bodies reciprocal which implies that the time dilation between them is also reciprocal, but the situation becomes more complicated for a body encompassing several segments of uniform motion and several bodies in a closed polygon in spacetime. In the case of the twins experiment, this polygon is a triangle, and each side of this triangle represents a different segment of uniform motion. The triangle has a reality in the spacetime of the relativity theory, which implies that the outgoing astronaut coexists with the returning astronaut in spacetime. There is therefore no reason to neglect the time dilation existing between them. Precisely the astronaut A R (after the U-turn), has a virtual relative velocity of v ⊕ v with respect to A O . With reference to the proper time of A O , the time dilations of the two other sides of the triangle are

∆t mov BA O ∆t A O = 1 1 -β 2 (16a) ∆t mov A R A O ∆t A O = 1 1 - (v ⊕ v) 2 c 2 = 1 + β 2 1 -β 2 (16b)
while from the point of view of B at rest

∆t mov A O B ∆t B = 1 1 -β 2 (16c) ∆t mov A R B ∆t B = 1 1 -β 2 (16d)
The asymmetry of the twin experiment can thus be read through the time dilations, provided that the time dilation hidden in the spacetime between A O and A R is taken into account. But contrary to the Lorentz transformations, relative time dilations do not form a group. The derivation of a differential aging from these relations would require some "law of composition of time dilations" to be defined. In its absence, it is risky to try to solve the twins' experiment by the unjustifiable application of a time dilation only to one of the twins.

Conclusions

Although it is often discredited, the twin experiment might be one of the most important questions of special relativity. When the defenders of the official solution oppose to their detractors that the twin paradox is a paradox only if special relativity is not understood, their detractors answer that it is rather the official solution which is a paradox for the special relativity theory. The formal treatment of time dilation in this theory shows that it is perfectly symmetrical, just like Lorentz transformations, and incapable of generating differential aging. The analysis of Minkowski spacetime diagrams can be misleading because of a frequent confusion between time dilation and graph scale dilation, and the time dilation depends on the arbitrary choice of the body at rest. In spite of their simplicity, light clocks allow to solve inambiguously the twins experiment. The comparison of the numbers of ticks of the ground clock and the traveling clock, gives a result independent of the choice of the point of view at rest and clearly demonstrates a break of temporal symmetry that cannot be grasped from the simple time dilation of special relativity. The surplus of different solutions given to the twins experiment has been pertinently made doubtful of its real resolution [START_REF]The Twin Paradox[END_REF]. It is suggested here that the time dilation based solutions should be avoided and thoses based on simultaneity lines and light clocks are much preferred. It turns out that the ratio of durations in the twins' experiment corresponds precisely to the time dilation factor γ. This coincidence was probably responsible for a large part of the objections to this result by researchers who saw in it a confusion between relative time dilation and absolute differential aging. Another factor may have contributed to this misunderstanding : the fact that this result was announced by Einstein in 1905 in his founding article of special relativity which essentially dealt with symmetrical time dilation. Einstein's lack of explanation on this point in this article and his subsequent hesitation in his later publications is surprising, especially when he later mentioned a possible role for acceleration through general relativity, even though he had not yet developed this theory in 1905. It cannot be excluded that Einstein had a flash of insight when he predicted this result in his 1905 paper without formalizing it, so that he could have forgotten it later. His foresight was also remarkable when he suggested, again without mathematical formalization, that the topology of the journey should not really matter, writing : "It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide. If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result : If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be 1 2 t v 2 c 2 second slow". The experiment of half-life of muons in circular trajectory gave him (once more) reason [START_REF] Sherwin | Some recent experimental tests of the "clock paradox[END_REF][START_REF] Bailey | Measurements of relativistic time dilatation for positive and negative muons in a circular orbit[END_REF]. In any case, as his unfalsifiable light-tube clock proves, an absolute time shift does exist in the round-trip experiment. This shift has little to do with the reciprocal time dilation of special relativity and conversely, the common use of the twin paradox to illustrate time dilation is nonsense.

Appendices

A Validation of the method of the mobile observation frame

In order to validate the method of the mobile observation frame abundantly used in this study, we will show that it allows to establish without complicated calculations major results of special relativity : the relative masses and energies. The limit velocity c and the relativistic addition of the velocities impose to play on the masses to maintain the conservation of momenta. The top panel of Fig. A1 is a possible representation of two balls of identical invariant mass, moving relative to each other at constant velocity v. We will show that the relation between the masses imposed by the relativistic addition of the velocities, is

m v = m 0 1 - v 2 c 2
To simply demonstrate this result, consider the observation of the same scene through a moving frame U whose uniform translation velocity is precisely adjusted to equalize the velocities and masses of the two balls (Fig. A1). The left ball, previously stationary, acquires a velocity u when seen through the moving frame while the right ball has its velocity modulus reduced from v to u (Eq.( 2) in the main text). Although u is no longer the relative velocity between the two balls, this new representation does not change the scene. The conservation of global mass imposes

m 0 + m v = 2m (A.1a)
and the conservation of momentum imposes

m v v = 2m u (A.1b)
The solution of this system of two equations depends on the rule used for the addition of the velocities.

• with a Galilean addition of velocities v = 2u, we find the Newtonian result m 0 = m v • while using the relativistic velocity addition

v = 2u 1 + u 2 c 2 gives m v = m 0 1 + u 2 c 2 1 - u 2 c 2 = m 0 1 - v 2 c 2 (A.2)
Once this formula is established, the other famous relations of special relativity come naturally. As it will be useful for the following appendix, we can for instance deduce the relativistic kinetic energy E K = v dp :

E K = v u=0 u d(mu) = v u=0 u d du   m 0 u 1 -u 2 c 2   du = v u=0 u m 0 1 -u 2 c 2 3/2 du =   m 0 c 2 1 -u 2 c 2   v 0 = m 0 c 2 1 -v 2

B Geometric calculation of the numbers of time units

The calculations can be done by simple Euclidean geometry after point-to-point projection of the hyperbolic Minkowski diagram onto the Euclidean plane on which it is drawn. To avoid any confusion with the relativistic interpretation of the diagram, the coordinate ct will be noted y as usual for an ordinary plane. The Euclidean distances on this plane have no spacetime significance but their ratios will allow to count the light tube time units (tick-tock-tick sequences contained in each inertial section). The principle of counting the clock beats in each participant in the experiment are described in Fig. B1. L(x) is the length of the tube seen at rest but two other lengths appear on this diagram. L (x ) is the length of this tube in motion with respect to the x coordinate. On the diagram L (x ) = L(x)

1 + β 2 1 -β 2 but in units normalized using the scale factor, L (x ) = L(x). The contraction of the lengths with respect to the reference frame at rest gives L (x) = L(x) 1 -β 2 .

B.1 Distances between the ticks

Let us first calculate the distances between two successive ticks, starting with the traditional diagram from the point of view of the inert observer (Fig. 6A). Since light rays can only take paths at 45 • , for the stationary tube, the tick-tock and tock-tick paths are necessarily equal and the distance between two ticks is 2L. For the moving tube, the tick'-tock' segment appears longer than the tock'-tick' one (Fig. B1). The coordinates of the starting tick' are x 0 = y 0 = 0, the coordinates of the tock' are given at the intersection between the light path y = x and the front of the tube y =

x -L (x) β , which gives x tock' = y tock' = L (x) 1 -β .

Finally the coordinates of the second tick are given at the intersection between the back of the tube y = x β and the line of the tock-tick radius y = 2 y tock' -x, from which we deduce x tick' = 2L (x)β 1 -β 2 and y tick' = 2L (x) 1 -β 2 . The distance between the points 0 and tick is therefore 0-tick' = (x tick' -x 0 ) 2 + (y tick' -y 0 ) 2 = 2L 1 + β 2 1 -β 2 where it only remains to replace L by L 1 -β 2 . These two intervals are sufficient for the diagram in Fig. 6A, where 0-tick' is valid for both sections ab and bc because their relative velocities with respect to the observer are indeed β even if their vector orientation is opposite. Hence,

Figure 1 .

 1 Figure 1. (A) Diagram of the round trip of the astronaut starting from B and returning to B after an elastic rebound on C. The planets B and C are considered as belonging to the same reference frame. The events a and c correspond respectively to the departure and the return of the astronaut. R is the turnaround point. (B) Authorized spacetime diagram represented in the rest frame of B. (C) Forbidden spacetime diagram represented from the rest frame of A.

Figure 3 .

 3 Figure 3. Einstein's light tube. (A) For the observer standing on the cart, the tube is immobile and the light travels exactly the length of the tube. (B) For the observer outside the carriage who sees the tube moving from left to right at velocity v, the path of the light traveling along with the carriage is oblique and longer than the length of the tube. (C) The points of view of the two observers are combined on a single diagram.

Figure 4 .

 4 Figure 4. Longitudinal light tube. The carriage appearing either (A) stationary, or (B) in uniform motion from left to right at speed v.

Figure 5 .

 5 Figure 5. Euclidean and relativistic motions associated respectively with an infinite and a finite speed of light. (A) Scheme of uniform longitudinal motion of the Newtonian light tube. (B) Revision of the Newtonian scheme imposed by a finite speed of light, identical for an observer either stationary (frame (ct, x)) or following the motion of the light tube (frame (ct , x )). In Minkowski representation, whatever the point of view, the paths of the light (red dotted lines) can only be diagonal lines at 45 • . The ratio of the lengths L and L describes the length contraction of special relativity and the ratio of the distances tick-tock-tick and tic'-tac'-tic' reflects the time dilation of special relativity. The diagram is drawn for β=0.6.

2 - 2 = cos 2

 222 (∆x cos arctan β + c∆t sin arctan β) arctan β -sin 2 arctan β ∆s 2 (11a)

Figure 6 .

 6 Figure 6. Triangles of the twins experiment, represented at the same scale in Minkowski diagrams from different points of view. (A) Most commonly used diagram, drawn from the resting frame of reference of the twin on earth. (B) Frame of reference of the traveler at rest during his outward journey. (C) Rest frame of the traveler during his return trip. (D) Rest frame of a traveler outside the experiment, slower than the traveling twin and always located halfway between the earth and the traveling twin during his outward trip. The diagrams are drawn for β=0.6 between the twins.

Figure 7 .

 7 Figure 7. Different spacetime diagrams of the twin experiment. (A,B) The rest frames of then astronaut are arbitrarily divided into two sections, one for the outward journey on panel A and the other for the return journey on panel B. A O is the astronaut during the outward trip and A R is the astronaut during the return trip. The diagrams are drawn from the rest frames of an observer located halfway between the travelling and non-traveling twins at speed u relative to the inert twin B, called U in panel A and Ũ in panel B (see Fig.2). (C) U and A continue straight on, instead of turning. Diagram seen from the rest frame U of located halfway between B and A. (D) Standard twin experiment in which A turns around and reaches the earth Diagram drawn from the rest frame of U who continues straight on without turning back.

Figure 8 .

 8 Figure 8. The two inverted round trips of the astronaut A 1 starting from B and returning to B after an elastic rebound on C and of the astronaut A 2 starting from C and returning to C after an elastic rebound on B. The planets B and C are considered as belonging to the same reference frame.

Figure A1 .

 A1 Figure A1. Thought experiment of the moving observation frame to find the value of the moving mass. In the top panel, the frame is fixed in the frame of reference of the ball of mass m0, in which the other ball of mass (mv) is moving at velocity v towards m0. v is the relative velocity between the two balls. It could as well have been attributed to the left ball if the observation frame had been locked on the right ball. In the bottom panel, the top scene is viewed through the frame in uniform motion to the left so as to give identical velocity modules to the two balls.

c 2 -m 0 c 2 (

 22

•N

  Figure B1. Minkowski diagram detailed for the calculation used here, describing the uniform motion of a tube of length L at rest. The hyperbola H for time normalization has the equation (ct) 2 -x 2 = 1. The unit of time of a light clock of length L arbitrarily set to a unit x = 1, is 2 ct at rest. The diagram is drawn for β=0.6.