Statistical Field Theory An Introduction To Exactly Solved Models In Statistical Physics 2nd Edition Giuseppe Mussardo - Solutions

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A report by the Gallup organization in February 2015 estimated that 9% of Republicans approved of President Barack Obama’s performance, whereas 79% of Democrats approved.Would you characterize the association between political party affiliation and opinion about Obama’s performance as weak, or
Table 8.23, from the 2014 General Social Survey, cross-classifies happiness and marital status.(a) Software reports that X2 = 135.3.Interpret.(b) The table also shows, in parentheses, the standardized residuals. Summarize the association by indicating which marital statuses have strong evidence of
For a 2×4 cross-classification of gender and religiosity(very, moderately, slightly, not at all) for recent GSS data, the standardized residual was 3.2 for females who are very religious, −3.2 for males who are very religious,−3.5 for females who are not at all religious, and 3.5 for males who
Table 8.22 shows SPSS analyses with the 2014 General Social Survey, for variables party ID and race.(a) Report the expected frequency for the first cell, and show how SPSS obtained it.(b) Test the hypothesis of independence. Report the test statistic and P-value, and interpret.(c) Use the
The sample in Table 8.16 is 382 black Americans.Table 8.21 shows cell counts and standardized residuals for income and happiness for white subjects in the 2014 General Social Survey.(a) Interpret the Pearson chi-squared statistic and its P-value.(b) Interpret the standardized residuals in the four
In the GSS, subjects who were married were asked the happiness of their marriage, the variable coded as HAPMAR.(a) Go to sda.berkeley.edu/GSS/ and construct a contingency table for 2014 relating HAPMAR to family income measured as (above average, average, below average), by entering FINRELA(r: 1-2;
Are people happier who believe in life after death?Go to the GSS website sda.berkeley.edu/GSS and download the contingency table for the 2014 survey relating happiness and whether you believe in life after death(variables HAPPY and POSTLIFE, with YEAR(2014) in the selection filter).(a) State a
Table 8.20 is based on results described in a study5 that examined the effects of alcohol consumption and drug use on sexual behavior, for undergraduate students at a university in the southeastern United States. In this table, the columns refer to whether the student engaged in unprotected sex in
In 2010, the GSS asked about willingness to accept cuts in the standard of living to help the environment, with categories (very willing, fairly willing, neither willing nor unwilling, not very willing, not at all willing). When this was cross tabulated with sex, X2 = 3.3.(a) What are the
A sociologist uses a 2 × 4 contingency table to compare four groups on a binary response variable. For group i, let πi denote the population proportion of response in the first outcome category, i = 1, 2, 3, 4. Explain what df = 3 means in the context of comparing these four population
How large a X2 value provides a P-value of 0.05 for testing independence for the following table dimensions?(a) 2 × 2 (b) 3 × 3 (c) 2 × 5 (d) 5 × 5(e) 3 × 9
Data posted at the FBI website (www.fbi.gov) indicated that of all blacks slain in 2013, 92% were slain by blacks, and of all whites slain in 2013, 93% were slain by whites. Let y denote race of victim and x denote race of murderer.(a) Which conditional distributions do these statistics refer to,
Based on current estimates of how well mammograms detect breast cancer, Table 8.19 shows what to expect for 100,000 adult women over the age of 40 in terms of whether a woman has breast cancer and whether a mammogram gives a positive result (i.e., indicates that the woman has breast cancer).(a)
TheWorld Values Survey4 asked, “How often do you pray?” The response never was given by 16.5% of the 2232 respondents in the United States, by 48.4% of the 1189 respondents in Spain, and by 56.8% of the 1206 respondents in Sweden. Show how to construct a contingency table relating the outcome
A March 2015 survey by the Pew Research Center(www.pewresearch.org) compared various groups of Americans in terms of their support for legalizing marijuana.In considering age groups, it estimated that legalization was supported by 68% of those of age between 18 and 34, by 51% of those of age
Whether a woman becomes pregnant in the next year is a categorical variable with categories (yes, no), and whether she and her partner use contraceptives is another categorical variable with categories (yes, no). Would you expect these variables to be statistically independent, or associated?
GSS surveys routinely show that in the United States, about 40% of males and 40% of females believe that a woman should be able to get an abortion if she wants it for any reason.(a) Construct a contingency table showing the conditional distribution on whether unrestricted abortion should be legal
For (a)–(c), is it a null hypothesis, or an alternative hypothesis?(a) In Canada, the proportion of adults who favor legalized gambling equals 0.50.(b) The proportion of all Canadian college students who are regular smokers now is less than 0.20 (the value it was 10 years ago).(c) The mean IQ of
You want to know whether adults in your country think the ideal number of children is equal to 2, or higher or lower than that.(a) Define notation and state the null and alternative hypotheses for studying this.(b) For responses in a recent GSS to the question “What do you think is the ideal
For a test of H0: μ = 0 against Ha: μ = 0 with n = 1000, the t test statistic equals 1.04.(a) Find the P-value, and interpret it.(b) Suppose t = −2.50 rather than
Find the P-value.Does this provide stronger, or weaker, evidence against the null hypothesis? Explain.(c) When t = 1.04, find the P-value for (i) Ha: μ > 0,(ii) Ha:μ < 0.
The P-value for a test about a mean with n = 25 is P = 0.05.(a) Find the t test statistic value that has this P-value for (i) Ha: μ = 0, (ii) Ha:μ > 0, (iii) Ha:μ < 0.(b) Does this P-value provide stronger, or weaker, evidence against the null hypothesis than P = 0.01? Explain.
Find and interpret the P-value for testingH0: μ = 100 against H0: μ = 100 if a sample has(a) n = 400, ¯y = 103, and s = 40.(b) n = 1600, ¯y = 103, and s = 40. Comment on the effect of n on the results of a significance test.
Example 6.4 (page 148) described a study about therapies for teenage girls suffering from anorexia. For the 17 girls who received the family therapy, the changes in weight were 11, 11, 6, 9, 14, −3, 0, 7, 22, −5, −4, 13, 13, 9, 4, 6, 11.Some Stata output (edited) for the data
According to a union agreement, the mean income for all senior-level workers in a large service company equals $500 per week. A representative of a women’s group decides to analyze whether the mean income μ for female employees matches this norm. For a random sample of nine female employees, ¯y
By law, an industrial plant can discharge no more than 500 gallons of waste water per hour, on the average, into a neighboring lake. Based on other infractions they have noticed, an environmental action group believes this limit is being exceeded. Monitoring the plant is expensive, and a random
In response to the statement “A preschool child is likely to suffer if his or her mother works,” the response categories (Strongly agree, Agree, Disagree, Strongly disagree)had counts (91, 385, 421, 99) for responses in a General Social Survey. To treat this ordinal variable as quantitative, we
In Example 6.2 on political ideology (page 145), suppose we use the scores (−3, −2, −1, 0, 1, 2, 3) instead of(1, 2, 3, 4, 5, 6, 7). We then test H0: μ = 0. Explain the effect of the change in scores on (a) the sample mean and standard deviation, (b) the test statistic, (c) the P-value and
Results of 99% confidence intervals for means are consistent with results of two-sided tests with which α-level? Explain the connection.
For a test of H0: π = 0.50, the z test statistic equals 1.04.(a) Find the P-value for Ha:π > 0.50.(b) Find the P-value for Ha: π = 0.50.(c) Find the P-value for Ha:π < 0.50.(d) Do any of the P-values in (a), (b), or (c) give strong evidence against H0? Explain.
For a test of H0: π = 0.50, the sample proportion is 0.35 with n = 100.(a) Show that the test statistic is z = −3.0.(b) Find and interpret the P-value for Ha:π < 0.50.(c) For a significance level of α = 0.05, what decision do you make?(d) If the decision in (c) was in error, what type of error
Same-sex marriage was legalized across Canada by the Civil Marriage Act enacted in 2005. Is this supported by a majority, or a minority, of the Canadian population?In an Ipsos Global poll conducted for Reuter News in May 2013 of 1000 Canadians that asked whether legalization should stand or be
When the 2010 General Social Survey asked,“Would you be willing to pay much higher taxes in order to protect the environment?” 459 people answered yes and 626 answered no. Software shows the following results to analyze whether a majority or minority of Americans would answer
TheWorld Values Survey12 asked, “Indicate the importance in your life of religion.” Of the people sampled in the Netherlands, Stata reports the following output for the proportion who answered, “very important” or “rather important” (instead of “not very important” or “not at all
In the scientific test of astrology discussed in Example 6.9 (page 163), the astrologers were correct with 40 of their 116 predictions. Test H0: π = 1/3 against Ha: π >1/3. Find the P-value, make a decision using α = 0.05, and interpret.
The previous exercise analyzed whether astrologers could predict the correct personality chart for a given horoscope better than by random guessing. In the words of that study, what would be a (a) Type I error? (b) Type II error?
A mayoral election in Madison, Wisconsin, has two candidates. Exactly half the residents currently prefer each candidate.(a) For a random sample of 400 voters, 230 voted for a particular candidate. Are you willing to predict the winner?Why?(b) For a random sample of 40 voters, 23 voted for a
The authorship of an old document is in doubt.A historian hypothesizes that the author was a journalist named Jacalyn Levine. Upon a thorough investigation of Levine’s known works, it is observed that one unusual feature of her writing was that she consistently began 6% of her sentences with the
A multiple-choice test question has four possible responses. The question is difficult, with none of the four responses being obviously wrong, yet with only one correct answer. It first occurs on an exam taken by 400 students.Test whether more people answer the question correctly than would be
Example 6.4 (page 148) tested a therapy for anorexia, using H0: μ = 0 and Ha:μ > 0 about the population mean weight change.(a) In the words of that example, what is a (i) Type I error? (ii) Type II error?(b) The P-value was 0.018.If the decision for α = 0.05 were in error, what type of error is
Jones and Smith separately conduct studies to test H0: μ = 500 against H0: μ = 500, each with n = 1000.Jones gets ¯y = 519.5, with se = 10.0.Smith gets ¯y = 519.7, with se = 10.0.(a) Show that t = 1.95 and P-value = 0.051 for Jones.Show that t = 1.97 and P-value = 0.049 for Smith.(b) Using α
Jones and Smith separately conduct studies to test H0: π = 0.50 against H0: π = 0.50, each with n =400. Jones gets πˆ = 220/400 = 0.550.Smith gets πˆ =219/400 = 0.5475.(a) Show that z = 2.00 and P-value = 0.046 for Jones.Show that z = 1.90 and P-value = 0.057 for Smith.(b) Using α = 0.05,
A study considers whether the mean score μ on a college entrance exam for students in 2016 is any different from the mean of 500 for students in 1966. Test H0:μ = 500 against Ha: μ = 500 if for a nationwide random sample of 10,000 students who took the exam in 2016,¯y = 497 and s = 100. Show
Areport by the Collaborative onAcademic Careers in Higher Education indicated that there is a notable gap between female and male academics in their confidence that tenure rules are clear, with men feeling more confident.The 4500 faculty members in the survey were asked to evaluate policies on a
Refer to Example 6.8 on “medical discoveries”(page 161). Using a tree diagram, approximate P(Type I error) under the assumption that a true effect exists 20%of the time and that P(Type II error) = 0.30.
A decision is planned in a test of H0: μ = 0 against Ha: μ > 0, using α = 0.05.If μ = 5, P(Type II error) =0.17.(a) Explain the meaning of this last sentence.(b) If the test used α = 0.01, would P(Type II error) be less than, equal to, or greater than 0.17? Explain.(c) If μ = 10, would P(Type
Let π denote the proportion of schizophrenics who respond positively to treatment.Atest is conducted ofH0:π = 0.50 against Ha:π > 0.50, for a sample of size 25, usingα = 0.05.(a) Find the region of sample proportion values for which H0 is rejected.(b) Suppose that π = 0.60.Find P(Type II
A follow-up to the anorexia study of Example 6.4 plans a test of H0: μ = 0 against Ha: μ > 0, for the mean weight change μ for a new therapy, predicting that μ = 10.The test uses α = 0.05.For n = 30, suppose the standard deviation is 18. Find P(Type II error) if μ = 10 by showing(a) a test
Refer to the previous exercise.(a) Find P(Type II error) if μ = 5. How does P(Type II error) depend on the value of μ?(b) Find P(Type II error) if μ = 10 and α = 0.01.How does P(Type II error) depend on α?(c) How does P(Type II error) depend on n?
A jury list contains the names of all individuals who may be called for jury duty. The proportion of women on the list is 0.53.A jury of size 12 is selected at random from the list. None selected are women.(a) Find the probability of selecting 0 women.(b) Test the hypothesis that the selections are
A person claiming to possess extrasensory perception(ESP) says she can guess more often than not the outcome of a flip of a balanced coin in another room, not visible to her.(a) Introduce appropriate notation, and state hypotheses for testing her claim.(b) Of five coin flips, she guesses the
In aCNNexit poll of 1336 voters in the 2006 Senatorial election in NewYork State, let x =number in exit poll who voted for the Democratic candidate, Hillary Clinton.(a) Explain why this scenario would seem to satisfy the three conditions needed to use the binomial distribution.(b) If the population
In a given year in the United States, the probability of death in a motor vehicle accident equals 0.0001 for females and 0.0002 for males (Statistical Abstract of the United States).(a) In a city having 1 million females, find the mean and standard deviation of x = number of deaths from motor
You can use the Errors and Power applet at www.artofstat.com/webapps.html to investigate the performance of significance tests, to illustrate their long-run behavior when used for many samples. For significance tests, set the null hypothesis as H0: π = 0.50 for a one-sided test withHa:π > 0.50
Refer to the Students data file (Exercise 1.11 on page 9).(a) Test whether the population mean political ideology differs from 4.0.Report the P-value, and interpret.(b) Test whether the proportion favoring legalized abortion equals, or differs, from 0.50.Report the P-value, and interpret.
Refer to the data file your class created in Exercise 1.12 (page 10). For variables chosen by your instructor, state a research question and conduct inferential statistical analyses. Also, use graphical and numerical methods presented earlier in this text to describe the data and, if necessary, to
A study considered the effects of a special class designed to improve children’s verbal skills. Each child took a verbal skills test before and after attending the class for three weeks. Let y = second exam score − first exam score. The scores on y for a random sample of four children having
The 49 students in a class at the University of Florida made blinded evaluations of pairs of cola drinks. For the 49 comparisons of Coke and Pepsi, Coke was preferred 29 times. In the population that this sample represents, is this strong evidence that a majority prefers one of the drinks?Refer to
The U.S. Justice Department and other groups have studied possible abuse by police officers in their treatment of minorities. One study, conducted by the American Civil Liberties Union, analyzed whether African-American drivers were more likely than others in the population to be targeted by police
An experiment with 26 students in an Israeli classroom consisted of giving everyone a lottery ticket and then later asking if they would be willing to exchange their ticket for another one, plus a small monetary incentive.Only 7 students agreed to the exchange. In a separate experiment, 31 students
Ideally, results of a statistical analysis should not depend greatly on a single observation. To check this, it’s a good idea to conduct a sensitivity study: Redo the analysis after deleting an outlier from the data set or changing its value to a more typical value, and check whether results
In making a decision in a test, a researcher worries about the possibility of rejecting H0 when it is actually true. Explain how to control the probability of this type of error.
Consider the analogy between making a decision in a test and making a decision about the innocence or guilt of a defendant in a criminal trial.(a) Explain what Type I and Type II errors are in the trial.(b) Explain intuitively why decreasing P(Type I error) increases P(Type II error).(c) Defendants
Medical tests for diagnosing conditions such as breast cancer are fallible, just like decisions in significance tests. Identify (H0 true, H0 false) with disease (absent, present), and (Reject H0, Do not reject H0) with diagnostic test (positive, negative), where a positive diagnosis means that the
An article in a sociology journal that deals with changes in religious beliefs over time states, “For these subjects, the difference in their mean responses on the scale of religiosity between age 16 and the current survey was significant (P < 0.05).”(a) Explain what it means for the result to
An article in a political science journal states that“no significant difference was found between men and women in their voting rates (P = 0.63).” Can we conclude that the population voting rates are identical for men and women? Explain.
You conduct a significance test using software. The output reports a P-value of 0.4173545. In summarizing your analyses in a research article, explain why it makes more sense to report P = 0.42 rather than P = 0.4173545.
A research study conducts 60 significance tests. Of these, three are significant at the 0.05 level. The authors write a report stressing only the three “significant”results, not mentioning the other 57 tests that were“not significant.” Explain what is misleading about their report.
Some journals have a policy of publishing research results only if they achieve statistical significance at the 0.05 α-level.(a) Explain the dangers of this.(b) When medical stories in the mass media report supposed large dangers or benefits of certain agents (e.g., coffee drinking, fiber in
We analyze whether the true mean discharge of wastewater per hour from an industrial plant exceeds the company claim of 1000 gallons. For the decision in the one-sided test using α = 0.05,(a) If the plant is not exceeding the limit, but actually μ =1000, there is only a 5% chance that we will
The P-value for testing H0: μ = 100 against Ha:μ = 100 is P = 0.001.This indicates that(a) There is strong evidence that μ = 100.(b) There is strong evidence that μ = 100.(c) There is strong evidence thatμ > 100.(d) There is strong evidence thatμ < 100.(e) If μ were equal to 100, it would
In the previous exercise, suppose the test statistic z = 3.29.(a) There is strong evidence that μ = 100.(b) There is strong evidence thatμ > 100.(c) There is strong evidence thatμ < 100.
A 95% confidence interval for μ is (96, 110).Which two statements about significance tests for the same data are correct?(a) In testing H0: μ = 100 against Ha: μ = 100, P > 0.05.(b) In testing H0: μ = 100 against Ha: μ = 100, P < 0.05.(c) In testing H0: μ = μ0 against Ha: μ = μ0, P >
Let β denote P(Type II error). For an α = 0.05-level test of H0: μ = 0 against Ha: μ > 0 with n = 30 observations,β = 0.36 at μ = 4. Then,(a) At μ = 5,β > 0.36.(b) If α = 0.01, then at μ = 4,β > 0.36.(c) If n = 50, then at μ = 4,β > 0.36.(d) The power of the test is 0.64 at μ = 4.(e)
Answer true or false for each of the following, and explain your answer:(a) P(Type II error) = 1 − P(Type I error).(b) If we reject H0 using α = 0.01, then we also reject it using α = 0.05.(c) The P-value is the probability that H0 is true. (Hint:Do we find probabilities about variables and
Explain the difference between one-sided and twosided alternative hypotheses, and explain how this affects calculation of the P-value.
Your friend plans to survey students in your college to study whether a majority feel that the legal age for drinking alcohol should be reduced. He has never studied statistics. How would you explain to him the concepts of (a) null and alternative hypotheses, (b) P-value, (c) α-level, (d) Type II
A random sample of size 40 has ¯y = 120. The Pvalue for testing H0: μ = 100 against Ha: μ = 100 is P = 0.057.Explain what is incorrect about each of the following interpretations of this P-value, and provide a proper interpretation.(a) The probability that the null hypothesis is correct equals
Refer to the previous exercise and the P-value of 0.057.(a) Explain why the P-value is the smallest α-level at which H0 can be rejected; that is, P equals the smallest level at which the data are significant.(b) Refer to the correspondence between results of confidence intervals and two-sided
Aresearcher conducts a significance test every time she analyzes a new data set. Over time, she conducts 100 tests.(a) Suppose H0 is true in every case. What is the distribution of the number of times she rejects H0 at the 0.05 level?(b) Suppose she rejects H0 in five of the tests. Is it plausible
Each year in Liverpool, New York, a public librarian estimates the mean number of times the books in that library have been checked out in the previous year. To do this, the librarian randomly samples computer records for 100 books and forms a 95% confidence interval for the mean. This has been
Refer to Example 5.8 on 0 vegetarians in a sample of n = 20 and Exercises 5.77 and 5.77.(a) At the Inference for a Proportion applet at www.artofstat.com/webapps.html, enter 0 successes for n = 20. Explain why the reported standard error values differ when you click on Confidence Interval and when
You test H0: π = 0.50 against Ha: π > 0.50, usingα = 0.05.In fact, Ha is true. Explain why P(Type II error) increases toward 0.95 as π moves down toward 0.50.(Assume n and α stay fixed. You may want to look at the applet introduced in Exercise 6.36.)
Refer to the ESP experiment in Exercise 6.33, with n = 5.(a) For what value(s) of x = number of correct guesses can you reject H0: π = 0.50 in favor of Ha:π > 0.50, usingα = 0.05?(b) For what value(s) of x can you reject H0 using α =0.01? (Note: For small samples, it may not be possible to
12.1. Vertex operators In order to prove that the ground states | α of the free bosonic theory are obtained by applying the vertex operators Vα(z,¯0) to the conformal vacuum | 0 it is necessary to show that Vα(0,0) is an eigenstate of π0 with eigenvalue α and, furthermore, that anVα(0,0) |
12.2. Modular transformations of the θi(τ ) functions.Let us denote by θi(z,τ) the theta functions of Jacobi, with θ1(τ ) = 0, while the others have the infinite series and product representationsθ2(τ ) =n∈Z q(n+1/2)2/2 = 2q1/8∞n=1(1−qn)(1+qn)2θ3(τ ) =n∈Z qn2/2
12.3. Dedekind function The Dedekind function is defined by the infinite product (q = e2iπτ )η(τ ) = q1/24∞n=1(1−qn).a. Use the definition of the θi(τ ) functions in terms of the infinite product to prove the identityη3(τ ) = 1 2θ2(τ ) θ3(τ ) θ4(τ ).b. Use the modular
12.4. Bosonic partition function at the self-dual point Consider the expression (12.2.56) of the partition function of a bosonic field with a compactification radius R equal to the self-dual value R =√2 Z(√2) = 1|η(τ )|2 n,m∈Z q14(n+m)2 ¯ q14(n−m)2 .Prove that this expression can be
12.5. Jacobi identity The aim of this exercise is to prove, by physical arguments, the Jacobi identity∞n=1(1−qn)(1+qn−1/2w)(1+qn−1/2w−1) =∞n=−∞qn2/2wn that holds for |q| < 1 and w = 0. Consider then the partition function of a free system of f fermions and ¯ f anti-fermions,
12.6. Quantum Pythagorean theorem A regularization of the normal order : A(x)B(x) : of two operators can be obtained by the limit limη→0 : A(x−η/2)B(x+η/2) : and an average on all directions of η, so that the final expression is invariant under the rotations. The average is equivalent to
12.7. Equivalence of the Sine–Gordon and Thirring models Consider the Sine–Gordon model of a scalar bosonic field ϕ, whose Lagrangian is L = 1 2(∂ϕ)2 + m2β2 (cosβϕ −1).Use the bosonization formulae, prove that this Lagrangian can be transformed in the Lagrangian of the Thirring model L
13.1. Spontaneous supersymmetry breaking Let Q be the generator of a N = 1 supersymmetric theory and Q† its adjoint operator.With a proper normalization we have{Q,Q†} = H, where H is the Hamiltonian of the system.a. Show that the Hamiltonian of a supersymmetric theory contains no negative
13.2. Central charge of the parafermions On a physical basis argue why in the limitN→∞the central charge of the parafermionic systems is equal to c = 2.
13.3. Polyakov–Wiegman identity Consider the action of the sigma model with a Wess-Zumino topological term S(g) = k 16πd2xTr(∂μg−1∂μg)+k.Prove the identity S(gh−1) = S(g)+S(h)+ k 2πd2xTr(g−1∂¯ zgh−1∂z h).Show that this identity gives rise to the invariance of the action
13.4. Correlation functions of the currents For the conformal models with a Kac–Moody algebra, compute the-point correlation functions of the analytic currents Ja(z1)Jb(z2)Jc(z3)Jd(z4).
13.5. Bosonization of the SU(2)1 theory Verify that the central charge of the theory SU(2)1 is c = 1. Compute the spectrum of the conformal weights of this theory and determine a representation of the corresponding conformal fields in terms of the vertex operators of a bosonic field ϕ.
14.1. Correlator of the Ising model Consider the following correlator of the two-dimensional Ising model H(η, ¯ η) = σ(∞)(1,1)(η, ¯ η)σ(0,0).Use the modified Coulomb gas to show that it is given by H(η, ¯ η) = 1 4η+1η1/2(1−η).
14.2. Structure constants Use the modified Coulomb gas to compute the correlation functions of the TIM.Determine the values of the structure constants given in the text.14.3. Vacua of the multi-critical Ising model Consider the potential of the multi-critical Ising model V(ϕ) = g1ϕ +g2ϕ2 +g3ϕ3
16.1. Backlund transformationsa. Write down the Backlund transformations for the Sine–Gordon model.b. Taking as initial solution of the equation of motion φ = 0, determine the new solution ˆ φ(z, ¯ z) and show that it coincides with the solitonic solution of the model.c. Iterate the procedure
16.2. Scattering processes of the solitons Analyse the solution (16.3.15) with topological charge T = 2 of the Sine–Gordon model in the limits t→±∞. Determine the time delay ss. Based on the positive sign of this quantity and the negative sign of the analogous quantity s¯s for the

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Statistical Field Theory An Introduction To Exactly Solved Models In Statistical Physics 2nd Edition Giuseppe Mussardo - Solutions

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